UFDC Home  myUFDC Home  Help 



Full Text  
NOVEL MITIGATION METHODS AGAINST REACTIVE JAMMING ATTACK: THEORETICAL AND PRACTICAL SOLUTIONS By INCHEOL SHIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 2010 Incheol Shin ACKNOWLEDGMENTS Most of all, I would like to acknowledge my chair advisor Dr. My Tra Thai. From the moment I started to work with her, she has encourage me, guided me through all the researches, and gave me invaluable advices, suggestions and supports to pursue this degree. I am heartily thankful to the members of my supervisory committee, Dr. Randy Y. C. Chow, Dr. Tamer Kahveci, Dr. Prabhat Mishra and Dr. Panos M. Pardalos, for their guidance and mentoring. Finally, I would like to show my gratitude to my family members. TABLE OF CONTENTS ACKNOWLEDGMENTS .................. LIST OF TABLES ...................... LIST OF FIGURES ..................... ABSTRACT .. .. .. .. .. .. . CHAPTER 1 INTRODUCTION ................... Reactive Jamming Attacks in WSNs Identification of Trigger Nodes .... Construction of Routing Backbone . Organization ............. 2 JAMMING ATTACKS ........... 2.1 Effectiveness of Jammers ..... 2.1.1 Packet Send Ratio (PSR) 2.1.2 Packet Delivery Ratio (PDR) 2.2 Jamming Attack Models ...... 2.2.1 Constant Jammer ...... 2.2.2 Random Jammer ...... 2.2.3 Deceptive Jammer ..... 2.2.4 Reactive Jammer ...... 2.3 Existing Solutions .. ....... 2.3.1 Physical Layer Approaches 2.3.2 Link Layer Approaches . 2.3.3 Network Layer Approaches 2.4 Conclusion .. ........... 3 CENTRALIZED IDENTIFICATION OF TRIGGER NODES ..... 3.1 Network Model and Problem Definition 3.2 Prelim inaries .. ............ 3.2.1 Maximum Clique Problem . 3.2.2 NonAdaptive Group Testing .. 3.3 Centralized Trigger Node Identification (CTNI) ....... 3.3.1 Group Victim Nodes Based on Minimum Collection Covers (GVNMCDDC) Algorithm .......... 3.3.2 Detection of Trigger Nodes Based on NonAdaptive Group Testing (DTNNCGT) Algorithm ....... 3.4 Theoretical Analysis .. .................. of Disjoint Disk Combinatorial page . 3 I I I I 1111 3.4.1 Estimation of Trigger Node Upper Bound D ... 45 3.4.2 Correctness .. ...... .. .. .......... ..47 3.4.3 Performance Analysis ......................... 48 3.4.4 Random Reactive Jamming Model ... 50 3.5 The TNLTCDS Routing Algorithm ... 51 3.6 Performance Evaluation ................. ......... 52 3.6.1 Simulation Setup ................. ......... 52 3.6.2 Results and Analysis ..... .. ..... 55 3.6.2.1 Performance by the number of jammers J ... 55 3.6.2.2 Performance by the number of radios m ... 55 3.6.2.3 Performance by the number of nodes N ... 56 3.6.2.4 Performance by the density of the network ... 56 3.6.2.5 Performance by transmission range of the jammers .. 57 3.7 C conclusion . .. 57 4 LOCALIZED IDENTIFICATION OF TRIGGER NODES ... 59 4.1 Network Model and Problem Definition .... ....... 59 4.2 Overview and Fundamental Results ..... 60 4.2.1 Overview of Identification Procedure ... 61 4.2.2 Hexagon Tiling Coloring ..... ... 62 4.2.3 The k Coloring Algorithm ....................... 63 4.3 Localized Trigger Node Identification (LTNI) ... 65 4.3.1 Partition of Nodes Based on Hexagon Tiling and Coloring 65 4.3.2 Trigger Nodes Detection Procedure ... 68 4.3.2.1 Sequential group testing based localized trigger node identification (SGTLTNI) algorithm ... 68 4.3.2.2 Identification of a single trigger node (ISTN) algorithm 69 4.4 Theoretical Analysis .. .. .. .. .. .. .. 70 4.4.1 Upperbound on Testing Rounds ..... 70 4.4.2 Message Complexity .......................... 72 4.4.3 Random Reactive Jamming Model ... 73 4.5 The TNLTCDS Routing Algorithm ... 74 4.6 Performance Evaluation ................. ......... 74 4.6.1 Testing Rounds T ....... ........ .. ...... .. 76 4.6.2 Message complexity .......................... 78 4.6.3 Runtime ..... ..... 81 4.6.4 The number of nodes in quarantine areas ... 83 4.6.5 Random reactive jammers ... 83 4.7 C conclusion . .. 84 5 CONSTRUCTION OF DOMINATING TREE ..... 86 5.1 Overview of Dominating Tree ......................... 86 5.2 Hardness and Approximation ..... ........ 87 5.2.1 Inapproximability ................. ......... 87 5.2.2 Approximating Dominating Tree .. .. 89 5.3 Heuristic Algorithm and Analysis ..... .... 91 5.3.1 Algorithm Description ......................... 92 5.3.2 Runtime Com plexity .......................... 94 5.4 Performance Evaluation ............................ 95 5.5 C conclusion . .. 99 6 CONSTRUCTION OF VIRTUAL BACKBONE WITH MULTIPLE FACTORS 100 6.1 Overview of Virtual Backbone ..... .. ... ... 100 6.2 Related W ork ..... . 102 6.2.1 General G raph . 102 6.2.2 Unit Disk G raph . 103 6.2.3 Disk Graphs with Bidirectional Links . 104 6.2.4 Other Results in CDS ......................... 104 6.3 Wireless Communication Model and Preliminaries .... 105 6.3.1 N stations . . 105 6.3.2 Term inologies . 105 6.3.3 Definitions ................... ............106 6.4 MultiFactors Model and Solutions .. .. 106 6.5 A Better Algorithm for CDS on Diameter .... 109 6.6 Progressive Algorithm (PA) and Analysis . ... 113 6.7 Further Improvements for The Progressive Algorithm .... 119 6.7.1 Reducing Multiple Paths ............... ....... 120 6.7.2 Removing Redundant Terminals . ... 121 6.7.3 Locating Central Area ................ ........ 123 6.8 Performance Evaluation ........................... 125 6.8.1 Performance for CDSBDD, CDSBD and PA .... 126 6.8.2 Performance for BDA and PA ..................... 128 6.8.3 Performance Based on Different ... 130 6.8.4 Performance for Improvement Techniques ... 131 6.8.5 Performance for MultiFactors Model ... 132 6.9 Conclusions .. .. .. .. .. ....... ... 133 7 CO NC LUSIO N . . 135 R EFER ENC ES . . 137 BIOGRAPHICAL SKETCH ................................ 145 LIST OF TABLES Table page 31 Notations ..................... ................. 38 61 R untim e(m s) . . 130 LIST OF FIGURES Figure page 21 Constant jam ming attack ................... ......... 19 22 Random jam ming attack ................... ............ 21 23 Deceptive jamming attack .......... ... ...... .......... 22 24 Reactive jamming attack .......... ... ..... ............. 23 31 Since item 6 (6th column) is a trigger node (positive item), only the 2nd and 6th groups (rows) return negative outcomes. On the contrary, all other four groups produce positive outcomes. . 44 32 5 Possible jammers activated by a trigger node t ... 46 33 Experimental results by various size of jammers ... 52 34 Experimental results by various size of channels ... 53 35 Experimental results by various size of nodes. . 53 36 Experimental results by various network densities ... 54 37 Experimental results by various size of a .. 54 41 The minimum distance between two nodes with same color ... 64 42 The coloring pattern for k = 4 . 65 43 Trigger nodes in a hexagon ............................. 71 44 Rounds by various parameters ..... ....... 76 45 Messages by various parameters ..... .. ..... 77 46 Runtime by various parameters ..... .. ...... 78 47 Nodes in quarantine areas by various parameters ... 79 48 The number of rounds T in random reactive jamming model with different values of jamming probability P. . 80 51 Reduction from WDS G to DT G' ... 89 52 An example of reduction from G to G' ...................... 90 53 The execution of HeurDT algorithm ..... .. .... 93 54 Simulation results for HeurDT, MSTL and optimal results ... 96 61 Given node r as the root, node a, b, f, g are terminals. r is the parent of c and e. Node a, b are siblings, f, g are siblings. Node a, b, f, g are r's 2hops away neighbors . . 106 62 An example for second phase in algorithm 12. .. 112 63 All the nodes in the ring are a CDS with diameter of 8 ... 117 64 An example for reducing multiple path ... 121 65 An example for removing redundant terminals . ... 122 66 An example of leaf nodes located at central area. The black nodes consist of the tree.................... .................. .. 124 67 An example of selecting the node with maximum degree as the root of CDS 125 68 Performance for CDSBD, CDSBDD and PA . ... 127 69 Performance for BDA and PA ............................ 128 610 Performance based on different ......... ....... ..130 611 Performance for the first and second improvement techniques 131 612 Performance for the third improvement techniques ... 132 613 Performance for multifactors model ..... .. .. .. 133 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NOVEL MITIGATION METHODS AGAINST REACTIVE JAMMING ATTACK: THEORETICAL AND PRACTICAL SOLUTIONS By Incheol Shin August 2010 Chair: My Tra Thai Major: Computer Engineering Wireless Sensor Networks (WSNs) consist of many spatially deployed sensor devices to cooperatively monitor selected target locations and communicate with base stations. WSNs have potential applications in various complex environments such as military surveillance, healthcare systems, disaster area monitoring and etc. Due to the requirement of largescale and realtime data processing in WSNs, security is critical in many sensor network applications. The wireless communication of sensor devices in resourceconstrained WSNs leads to many challenging and intriguing securitysensitive problems that cannot be handled using conventional security solutions. Especially, since jammers attack the broadcast nature of the transmission medium by injecting interference signals, they are considered the most critical and fatally adversarial threats to subvert or disrupt the networks. Jamming attacks do not have to modify communication packets or compromise any sensors in order to launch, which makes them difficult to detect and defend against. Existing countermeasures against such attacks either consider sensor nodes equipped with sophisticated hardware, which is infeasible for low cost resource constraint WSNs, or have high overhead in term of time and message complexity and energy consumption. However, to overcome the problems in existing methods against reactive jamming attack, this dissertation introduces a novel approach by identifying the trigger nodes, whose transmissions activate any jammers and then constructing routing path based on the identification of sensor nodes. This dissertation includes two main parts, identification of trigger nodes and construction of virtual backbone based on the detection. First of all, the identification of trigger nodes can help us (i) to design a better routing protocol by switching these nodes into only receivers to avoid activating jammers and (ii) to locate the jammers based on the trigger nodes, thus providing an alternative mechanism against reactive jamming attacks. Secondly, an effective construction of virtual backbone provides not only the jamming avoidance protocol but also efficient broadcast routing. For the adaptive building of virtual backbone, this dissertation investigates dominating tree and multifactor model studying a joint optimization problem on Connected Dominating Set (CDS) in wireless adhoc networks. Theoretical analysis and experimental results endorse the suitability of these identification algorithms in terms of time and message complexity. This work is the first one to design novel countermeasures against reactive jamming attack. CHAPTER 1 INTRODUCTION Wireless Sensor Networks (WSNs) have enormous mission critical applications in such different realms as military surveillance, disaster area alarm system and critical infrastructure protection monitoring. Hence, the quality of service and security are critical issues in WSNs which need an elite attention. Nodes in WSNs deliver packets by radio transmissions in a hopbyhop manner, and due to this broadcast advantage, a transmitter can communicate a message to all its receivers within it's transmission range by a single radio transmission. Broadcast nature, however, increases the vulnerability of WSNs to various security challenges mostly from Denial of Service (DoS) attacks as explained by [8]. Jamming attacks, one of the DoS attacks, especially are light weighted, but the most fatal threats to WSNs, because they attack the core of wireless broadcast advantage even without modifying communication packets and compromising wireless sensor devices. Due to the excessive hardware requirement on existing methods, threats cannot be neutralized and nullified with conventional security solutions. 1.1 Reactive Jamming Attacks in WSNs There are several types of Denial of Service (DoS) attacks, such as selective forwarding, sinkhole, Sybil attack, flooding and etc. However, jamming attacks are known as the most significant threats because of their effectiveness and lethal damages to WSNs. In jamming attacks, the malicious disseminates out adversarial signals into busy channels which are filled with legitimate sensor transmissions without following any legitimate protocols. This will result in the slump of the Signal to Noise Ratio (SNR) and network throughput. Furthermore, reactive jamming attack adopts more advanced and efficient jamming strategy than any other active jammers in which jammers quietly scan all the available channels to sense activities and blatantly start injecting adversarial signals on that channel to corrupt ongoing packets. Since it maintains lower energy consumption to launch but greater hardness to be detected, reactive jammers can cause more severe damages to WSNs. That is, the reactive jamming attack especially is the most intelligent and stealthy type of jamming attack. As the reactive jamming attack does not need any special hardware or a complicated mechanism to launch, it is simple to initiate and hard to be detected in WSNs. Importantly, it cannot be dealt with using conventional security mechanisms, such as cryptography, authentication and authorization, as it cannot prevent messages from being received by a malicious node if it is in range of the transmitter and operating at the same radio frequency. Consequently, this is particularly dangerous to WSNs since its main targets are the kernel characteristics of the networks, distributed largescale networks and the broadcast nature of wireless communication. Even though the existing literature has various solutions for different jamming models, the majority of them are not suitable to resource constrained networks and suffer from hardware requirements in wireless devices. In order to overcome the shortcomings from classic approaches, this dissertation introduces novel methods against the reactive jamming attack in WSNs by detecting the trigger nodes and then building jamming avoidance routing path according to the identification of wireless devices. Consequently, this dissertation consists of two parts, identification of trigger nodes and construction of virtual backbone procedure. 1.2 Identification of Trigger Nodes The identification of trigger nodes can have several benefits for mitigation of the attacks. First of all, routing algorithm can be constructed in which the triggers are only the receivers, thus avoiding activating the jammers and minimizing the adversarial effect of jamming attacks. This can overcome the limitations of excessive hardware requirement of the channel surfing and frequency hopping methods. In addition, the identification of trigger nodes will not create an unnecessary large size of jammed regions like in [78]. Secondly, in case trigger nodes needs to send messages, they may still utilize channel surfing, but only a few nodes may require this operation, thus greatly reducing the computational costs required by existing methods. That is, after the efficient identification of trigger nodes, victim nodes would be also scheduled to transmit messages in order to minimize the damage from the attackers by keeping silent during the transmission of the trigger nodes, thus preventing jamming. Finally, the location information of the trigger nodes to later locate the jammers within networks, which is the first attempt to provide location information in order to pinpoint the location of jammers. In addition, this study introduces not only centralized and localized mitigation methods but also random reactive jamming model in order to validate the faulttolerant identification approaches. The reactive jammers randomly react to legitimate ongoing communication activities with probability p < 1, which makes identification approach challenging. However, both centralized and localized approaches are still applicable against this models as well, so that their suitability can be verified in practical network environments. Even though this trigger identification is a nontrivial localized approach to be realized in practical network environments due to the unknown and dynamic locations of jammers with various behaviors, this study designs efficient centralized and localized mitigation schemes with low time and affordable message complexity. 1.3 Construction of Routing Backbone A WSN consists of sensor nodes communicating with each other using radio transmission. Nodes that are within their own transmission ranges can exchange messages directly, but nodes which are out of transmission range communicate through intermediate nodes to route their messages. Importantly, constructing virtual backbone with only nontrigger nodes after identification of all trigger nodes will help to maintain the throughput of networks against reactive jamming attack. Furthermore, due to the energy consumption of each node for transmissions, minimizing the size of intermediate nodes would help to reduce network energy consumption. In other words, since the sensor nodes in a WSN have limited resources and small transmission ranges, how to construct a set of intermediate nodes to relay messages determines the efficiency of routing in networks. Consequently, an efficient method of constructing a virtual backbone would be able to efficiently reduce not only the routing overhead of networks to maximize the network lifetime but also the size of jammed area to minimize damages from jamming attack. 1.4 Organization The rest of the work is organized as follows. Chapter 2 describes the evolution of jamming strategies to reactive jammers and existing countermeasure against the attacks. For the identification phase, Chapter 3 provides the solution for centralized identification of trigger nodes, meanwhile the localized algorithm for the detection of trigger nodes is presented in Chapter 4. Our proposed new problem of dominating tree studied in Chapter 5 followed by the multifactor model in Chapter 6 so as to provide the efficient construction of jamming resistant routing protocol. Chapter 7 concludes this dissertation. CHAPTER 2 JAMMING ATTACKS We new briefly introduce the jamming attacks and conventional security service mechanisms against them. Current jammers have been evolving and reacting against countermeasures using several strategies, including energy efficiency and lower probability of detection with higher stealth. Especially, some of them are equipped with protocolaware jamming attack models in order to achieve jamming effectiveness with lower energy consumption and higher stealth. In this chapter, various types of jamming attacks and existing solutions against them using detection mechanisms will be introduced in order to show how powerful jammers are with their evolving strategies. 2.1 Effectiveness of Jammers At present, there are two ways to measure the effectiveness of jamming attacks: Packet Send Ratio (PSR) and Packet Delivery Ratio (PDR). These two types of measurements are useful to quantify the efficiencies of jamming techniques. Research [52, 55, 69, 84, 86] describing jamming attacks based on these measurements. In this section, the importance of jamming attacks and how powerful they are will be introduced as they apply to PSR and PDR. 2.1.1 Packet Send Ratio (PSR) This is the ratio of the number of packets that are successfully transmitted as legitimate traffic compared to the volume of packets that were intended to be sent out. This is defined as: SentPackets PSR = IntendedToBeSentPackets In most MAC layers in WSNs, carrier sensing multiple access (CSMA) control has to be performed before legitimate communications in sender nodes. That is, sender devices are required to sense activity in the channel for a certain duration of time before transmission in order to avoid messages colliding. Jammers can drop the PSR by attacking the CSMA control protocol. A jammer can decrease the number of messages sent by injecting continuous interference signals into the idle channel. Since interference signals in the channel are sensed as busy in nodes that use the CSMA control protocol, senders have to hold the MAC layer message buffer without any message transmission until sensing the channel is idle. During holding, the message buffer fills with messages, and even new messages may keep arriving for the full buffer. A series of new messages could easily be dropped because the buffer is full. The worst scenario for the buffer would be a case when the existing messages keep aging during their waiting in a buffer of MAC layer, resulting in timeout and discard without transmission. In addition, the backoff time in each node would be drastically increased, and in CSMA protocol it takes considerable time to recover from the unnecessarily longer backoff time. The measurement of PSR can easily be implemented without any sophisticated hardware in the sensor devices since they only need to keep track of the number of messages from the upper layer and the size of the messages sent via legitimate traffic. Jammers attacking the PSR can be detected relatively easily by exploiting the Cyclic Redundancy Check (CRC) in the MAC layer. Each node would be monitoring the activities on the channel during adversarial messages from jammers, and if the adversarial messages cannot be validly checked through the CRC, then nodes would become suspicious of the presence of jammers in networks. Stealth is critical to attackers, and jammers may able to emit regular packets for higher stealth, but, the jammers need to acquire a way to generate legitimate packets, which depends on sophisticated hardware to analyze the protocol in order to disguise their packets as legitimate packets. 2.1.2 Packet Delivery Ratio (PDR) This is the ratio of the number of packets that are successfully validated through the CRC check compared to the size of packets received. The PDR is defined as: PacketsPassedCRC PDR = ReceivedPackets A Cyclic Redundancy Check (CRC) is a simple nonsecure hash function for detection of errors in communication messages. Usually the CRC computation would be done in the link layer by a long division operation where quotients are discarded and remainders from the operation becomes the results of the CRC, which are appended to the end of link layer frames. Receivers would be able to decode and validate received packets with ease by utilizing the CRC check. The PDR ratio drops if jammers inject interference signals into communication messages. A jammer would be able to decrease the number of packets that are successfully delivered to destination nodes by corrupting ongoing communication packets. Due to the broadcasting nature of wireless communications, collisions among packets are considered a critical problem, and the jammers attack the PDR from the viewpoint of this weakness by corrupting interference signals. Attacking the PDR not only drops the PDR, but also wastes the energy of sensor networks due to the fact that collided messages require retransmission. The sensor networks are resourcelimited networks, and energy consumption has been much studied because it dominates the network lifetime. In other words, invalidation of packets by CRC checking imply retransmissions in sender nodes after notifications from receivers, which wastes significantly energy and results in a shorter network lifetime. As a result, we can say that attacking the PDR from jammers causes more serious problems than attacking the PSR. In addition, in terms of stealth, it would be better for jammers to drop PDR. It would not be necessary for jammers attacking PDR to expose themselves to legitimate nodes when emitting their adversarial signal. That is, it is harder to defend the jamming attacks dropping the overall PDR of networks. 2.2 Jamming Attack Models There have been a wide range of jamming strategies to prevent legitimate users from transmitting or receiving communication messages. The objectives of the attacks are to effectively reduce PSR and PDR by emitting interference signals. The efficiency of attacks associated with PSR requires some knowledge about WSN communication protocols, while reducing PDR does not. We will discuss more about the various types of jammers according to their intelligence. 2.2.1 Constant Jammer Jammer Victim Node O Constantly generating interference si nal TransmissTransmissi S rangrange a /\n norr nodenode JAMMING 1^ \ .o.'"\ Figure 21. Constant jamming attackonstantly interference signal range ofa normal node Figure 21. Constant jamming attack According to the Fig. 21, the constant jammer, a trivial jammer, is the most intuitive strategy to implement, but it has the least energy efficiency among all jammers since it does not follow the protocols in WSNs, just emitting a series of noise signals into the channel. It can be implemented using a waveform generator, so as to emit continuous interference signals by disregarding the MAClayer protocol. The jammers can also be a normal wireless device used to transmit random bits without any consideration of protocols. While it does not require complicated monitoring architectures to sneak into communication protocols, doing so drastically hinders the performance of PSR since the most number of wireless networks exploit Carrier Sense Multiple Access / Collision Avoidance (CSMA/CA) to physically monitor the traffic at communication parties. That is, legitimate senders would not be able to initialize transmission procedures to send packets as long as there are signal detections including jamming signals and legitimate communication signals. As a result, the senders have to stall for a considerable length of time to avoid collisions, and PSR would drop drastically. This attack might work well in networks with high traffic. A constant jamming attack is ineffective in terms of energy consumption due to the fact that the constant jammers emit interference signals whether or not there is legitimate traffic. The jammers are also wireless devices that emit interference signals with limited resources, usually portable energy supplies, and demand high energy consumption to hold a communication channel in order to prevent nodes from sending packets. In addition, the stealth of adversaries cannot be effectively achieved due to the fact that their behavior disobeys to the protocols in networks. To maximize the stealth of jammers, they must be invisible to legitimate nodes under the communication protocols. The legitimate nodes can identify the presence of jammers with ease by monitoring signals from any nodes, including jammers, and wait until the jammers disappear. As a result, the constant jammer is easy for adversaries to launch, but hard to avoid detection from legitimate nodes. Figure 22. Random jamming attack 2.2.2 Random Jammer As depicted in Fig. 22, the random jammer has evolved from the constant jammer to conserve energy. Like the constant jammer, it emits jamming signals for a certain amount of the time, but sleeps after turning its radio off in order to conserve energy for a longer lifetime of jamming. While the switching mechanism between sleeping mode and jamming mode can save a significant amount of energy, it would drop the efficiency of attack to PSR. Because, during its jamming phase, normal senders would not be able to transmit messages for channel activities under jamming signals and would have to wait until its sleeping phase begins. They could send out the stalled messages during the jamming phase, and most of the messages would be successfully delivered without interference during the jammers' sleeping mode. This becomes a tradeoff between energy conservation, and accommodating an attack, which leads to another evolution of jamming strategy. 2.2.3 Deceptive Jammer Figure 23. Deceptive jamming attack By monitoring the protocols in WSNs, this is the first type of jammer that takes the stealth of jammers into consideration. The deceptive jammer in Fig. 23 does not send out random bits or waveforms by a generator, but regular packets in order to capture the channel of legitimate communicators. The deceptive jammers emitting regular packets forces legitimate communicators into the receiving state and prevents them from converting their state into send mode. This method was initially implemented by continuously sending out preamble messages, so that it is hard to detect and is an effective method to disable a CSMA. The deceptive jammer not only would drop the PSR, but also increase the stealth against the detection system by transmitting regular frames into a MAClayer channel. Legitimate sensor devices have no way to identify the packets from deceptive jammers without an authentication scheme, but an authentication procedure in the WSNs is expensive. In addition, normal nodes have no choice but wait until timeout from the receive mode when receiving the fake preamble messages from the jammers since the preamble messages indicate a constant stream of incoming packets. Continuously sending out regular preamble packets from the jammer would hurt PSR by holding the channel without any suspicion of the presence of jammers. As a result, the falling PSR causes a lower throughput int the networks. [2] also mentions a periodic jammer, another adaptation of deceptive jammer. Instead of sending out continuous preamble messages, it generates a series of short pulses in every DIFS interval (50 nanosec), so normal nodes find the network always busy. 2.2.4 Reactive Jammer STrigger Node Victim Node Jammer V5rV5 f. ,_:f /i ,(w:.' Monitoring communication Reactely V2 legitimate activities V gimerating Figure 24. Reactive jamming attack Reactive jamming attacks are considered one of the most intelligent jamming strategies due to their reactive behavior. The reactive jamming attack in Fig. 24 is one such DoS security threat, in which a malicious node jammerr) quietly scans all the available channels in a wireless network to sense any activity and, if it detects some signal from a legitimate node on any channel, it starts injecting Noise on that channel in order to interfere with all the receivers in its range resulting in a drastic decrease in the signal to noise ratio and communication throughput of the network. This kind of attack is straightforward to initiate and very effective as it has no special hardware requirements, so the jammer only needs to keep listening for a legitimate signal on the available radio channels. Reactive jamming attacks are able to effectively attack the PDR by using a reactive strategy. While attacking the PSR using the jammers introduced above would only be able to hold a channel away from normal nodes and waste the channel resource. Just dropping the PDR by using a reactive jammer inflicts more damage to networks than just dropping the PSR because of the retransmission scheme in legitimate senders. An intelligent jammer [2] with knowledge of the protocols is one of the implementations of a reactive jammer and it can have several different strategies to corrupt messages, including control messages and data messages. Importantly, it would be able to distinguish between control messages and data messages based on the length of each communication message or the intervals between them in WSNs. According to the jamming strategies, there could be four types of intelligent jamming attacks, such as CTS corruption jamming, Acknowledge corruption jamming, data corruption jamming, and DIFS wait jamming. *A clear to send (CTS) corruption jammer would wait until sensing an arbitrary request to send (RTS) frame from any normal nodes wishing to send data. The jammer initializes a short jamming pulse right after every end of an RTS packet during a period of SIFS since it knows that a CTS messages follows a RTS message as a next control message. After a period of SIFS, it will inject that short pulse into the shared medium in order to corrupt CTS messages with less energy consumption. As a result, the legitimate sender needs to reissue the RTS message to initialize communication since the CTS message did not get through. * An acknowledge corruption jammer disrupts acknowledgments after data packets, but this strategy requires message identification schemes to locate the ends of data packets, which means longer activation of the jammers in order to analyze packets. This is the least efficient jamming strategy among other intelligent jammers. A data corruption jamming attack strategy is relatively easier to launch than other strategies due to the fact that the data packet is longer than any other packets. However, it would drive poor PDR of WSNs due to the retransmission of data packets with additional communication procedures. A DIFS waiting jamming attack corrupts RTS packets or CSMA/CA data packets by monitoring the medium idle for DIFS time in networks with high traffic. 2.3 Existing Solutions There are many studies against jamming attacks [7, 26, 31, 32, 3537, 43, 47, 61, 70, 78, 83, 85, 87], but the high computational overhead of these methods badly reduces the effect in resourcelimited network environments, such as WSNs. For example, in the channel surfing methods from [7, 35, 37, 47, 78, 83, 85, 87] and frequency hopping methods from [26, 31, 32, 36, 43, 58, 61, 70], the transmission frequency or channel is changed to a range where there is no interference from the adversaries. These strategies are not quite suitable for WSNs, especially in multichannel WSNs, since the sensors have to scan all the channels to detect the jamming attacks and hop to new frequencies all the time, even in the middle of a communication. Due to the fact that most of the sensor nodes have a halfduplex transceiver on them, scanning the channels during transmission causes communication stalls to check the availability of the current channel. Frequent communication stalls result in a longer transmission duration and more energy consumption. Consequently, these methods cannot avoid high overhead and resource consumption. 2.3.1 Physical Layer Approaches Even though the spreadspectrum communication was introduced by the military for secure communications and resistance against natural interference, the use of spreadspectrum communication in the physical layer is one of the bestknown schemes to evade jamming attacks. In the literature, there are two ways to build a spreadspectrum communications against jamming attacks, frequency hopping (FH) [26, 31, 32, 36, 43, 58, 61, 70] and code division multiple access (CDMA) [14, 21, 30, 53, 76, 89], forms of direct sequence spreadspectrum communication. Since most jamming attacks are categorized into physical layer attacks, the beginning of the defense mechanisms start in the physical layer. This section will introduce existing physical layer countermeasures against the jamming attacks. The frequency hopping technique was designed as a kind of secure solution against jamming, eavesdropping, tempering, etc, in wireless communication. Jamming attacks are able to attack WSNs with partialband noise at the beginning stage of jamming attacks, and the frequency hopping solution is utilized to defend against the jammers with partialband noise at that time. The initial outline of the conventional FH method is to use each frequency slot to transmit packets through one of orthogonal signals during a certain period of time, signaling interval. That is, the transmitter hops between safe frequencies based on a predefined algorithms. However, the most important issue to exploit frequency hopping technique is how to secretly establish switching sequences between two communication parties in order to foil a third party. That is, reducing the probability of inference is the key of this technique against jamming attacks. A preshared secret code for FH is not feasible in wireless communication networks due to the dynamic behavior of sensor nodes and the scalability of the networks. Since the main issues of FH approach are how to assign the hopping sequences for shifting frequencies and how to synchronize them among the nodes, there has been much research regarding those critical issues in FH mechanisms. To remove the burden of distributing sequences, [61] proposed the uncoordinated frequency hopping (UFH) technique for an antijamming pointtopoint scheme to establish a secrete key between two communication parties. This paper introduced a UFHbased message transfer protocol for key establishment with an encapsulation technique for message fragments even in the presence of a jammer, and sensor nodes after the key establishment would be able to communicate each other with the coordinated frequency hopping method. The key establishment problem is inherent to the frequency hopping technique, but this paper tries to break the dependency between them. Importantly, the fragmentation procedure utilizes a collisionresistant hash function as well. Since this scheme is based on the uncoordinated hopping model, it suffers from lower communication throughput. That is, numerous sending attempts to transmit each fragment so that the performance of the scheme is greatly effected by the number of attempts. In addition, it incurs higher storage and processing costs in sensor nodes as well, which are too expensive for resource limited sensor networks. Recently, the messagedriven frequency hopping (MDFH) technique [42] has been introduced to achieve a more spectral efficiency than any other existing FH designs. In the traditional models, FH techniques require a much wider spread bandwidth than they actually use to transfer messages, which means the spectral efficiency from the total number of available carrier frequencies is too low in practical network environments. [42] presented an innovative form of MDFH that exploited message streams as a pseudorandom sequence for FH selection. The data stream is divided into multiple blocks, and each block consists of additional carrier bit vectors to determine the hopping frequencies and ordinary bit vectors that are actual data to transmit. A carrier bit vector has Bc to specify a hopping frequency to transfer the data and Nh the number of hops within a symbol period. The data blocks are fed into a serialtoparallel (S/P) converter in order to split two carrier bits and ordinary bits into two parallel data streams. In addition, this paper presented enhanced MDFH to utilize multiple transmissions at each hop by exploiting all the available carriers. This solution helps to increase the resistance against jamming attacks using an unpredictable messagedriven hopping pattern, which also removes the burden of synchronization somewhat between the communication parties. However, this frequency hopping technique also has a fundamental limitation concerning a follower jammer, also called repeater jamming attack. Since the follower jammers with determinator circuits attempt to determine the hop frequencies and generate jamming signals in a range of frequencies, importantly, the FH technique has a critical limitation against follower jamming attacks. Initially, the simplest solution for follower jammers was a technique using fast hopping in order to prevent the follower jammers from having sufficient time to determine the current frequency and emit interference signal on that. However, this approach was not feasible in resourcelimited network environments. Despite this, there have been several studies on the follower jamming attacks [26, 31, 32, 36, 70], no complete solution to the problem of follower jamming attacks has so far been reached. The frequency hopped mary frequency shift keyed (FH/MFSK) was designed as a countermeasure against a partialband noise jammer [36]. The implementation of FH/MFSK utilizes M nonoverlapping frequency synthesizers in each transmitter and receiver to hop each MFSK signal since the performance from conventional FH/MFSK implementation is greatly effected by the deviation of a single carrier. However, this approach requires complicated hardware to realize in a practical system, which is not infeasible for resourcelimited sensor networks. [31, 32] introduced a different approach as a countermeasure evolved from FH/MFSK against follower jammers. The multiple orthogonal signals during a certain time period, signaling interval, would be emitted in each frequency slot that is divided from total spread bandwidth. Furthermore, there is conventional mode and unconventional mode in order to synchronize communicators. In the case of conventional mode from pseudorandom probability of pc in both senders and receivers, a receiver would determine the transmitted data by the largest output via a dehopper. The unconventional mode from the probability of 1 pc does not require sending any information, but a receiver determines the data based on the presence of energy in the selected frequency slots. This approach might relieved the burden of the hardware requirement a little, but it still suffers from a synchronization problem between transmitters and receivers and lower spectral efficiency of total bandwidth. Direct Sequence Spread Spectrum (DSSS) is one implementation of spreadspectrum communications against jamming attacks in the physical layer by using narrow band jamming suppression capability. In this, data packets are modulated with a continuous and predefined chipping sequence, and the modulated signal is spread in frequency domain thus becoming resistant against the narrow band jamming signal. Initially, the wavelet packet modulated direct sequence Spread Spectrum (WPMDSSS) system [89] was designed to be immune to an interference signal, but this requires sophisticated modulating hardware in the sensors. The main issue of the DSSS technique is also secret key sharing due to the fact that the performance of suppression capability is limited by the pseudo random generator with the keys among communicators. The key establishment to drive identical spreading codes among the communicators is a critical issue for the wireless adhoc networks in terms of scalability. The secrets for the DSSS communication between the sender and the receivers have to be in agreement before the start of legitimate communications, but this preshared secret key has been considered a difficult problem to solve due to the dynamic behavior of sensors. [53] proposed a solution called Uncoordinated DSSS (UDSSS) for authentic spread spectrum antijamming communication in order to solve the key sharing problem. This approach enables the implementation of DSSS without prior establishment of secret keys among the communicators, and receivers keep a certificate of the sender's public key instead of sharing secret keys. The transmitters do not use predefined spreading code, but randomly choose one of the codes that are available in public in order to prevent any receivers from inferring a choice of transmitter. The receivers using the UDSSS scheme despread received messages by applying each sequence out of the set using a trialanderror method. The networks they assume are not time synchronized, so that transmitters have to send out a message with repetition, and receivers synchronize with senders by applying a sliding window approach. That is, the time to identify a correct spreading code and synchronization procedure in receivers dominate the overall performance in the UDSSS method, and it has less efficiency than the DSSS method. With respect to the last approach CDMA among spreadspectrum communications, one study [14] related to utilizing the direct sequence CDMA (DSCDMA), resorts to highpower dynamic treeremerging schemes to maintain the small number of orthogonal codes in use and to avoid recalculation of the codes. Spread spectrum communication has been studied to resist jamming attacks in unicast communication because the number of codes greatly effects the performance of DSSS. [14] developed a broadcasting technique with a DSSS scheme with fewer number of codes by using a binary key tree. Due to the variation in nodes in dynamic environments, this method suffers from the additional maintenance overhead of joinin and leaving behaviors, especially, so computation of orthogonal codes takes much time. In addition, because of the probabilistic nature of packet reception, it also has a problem of false alarms, and as a solution for false alarms, networks are periodically required to reset their code in each sensor and build the code tree repeatedly. This series of reconstruction procedures necessitates high message complexity and additional computation overhead on the networks as well. In these respects, a scheme with cryptographic key management has a scalability problem and a stability problem when applied to various dynamic networks. As explained above, the existing FH methods have a fundamental problem with follower jammers when a determinator circuit attempts to determine the hop frequency and generate jamming signals in a range of the frequencies. DSSS has limitations against attacks based on Radio Frequency Memory (RFM) [30] as well since the repeater jammers try to acquire the code by monitoring the ongoing traffic and garble the communication messages based on the acquisition of the code. [76] have analyzed DSSS techniques attacked by repeater jammers in detail, but so far there has been no complete solution for repeater jamming attack. Due to the nature of a jamming attack, most defense schemes start from the physical layer, FH and DSSS. FH methods from [26, 31, 32, 36, 43, 58, 61, 70] hop communication frequencies seeking a safe one in order to avoid the jammed communications based on the switching sequences. How to distribute the hopping sequence and synchronize communication partners with the new sequences are the main issues to overcome low spectral efficiency. Code Division Multiple Access (CDMA) scheme communication [14, 21, 30, 53, 76, 89], a form of direct sequence spread spectrum is also one of the most common way to resist jamming attacks. However, the problem of how to manage secret keys for efficient suppression capability has to be solved for better performance of immunity against attack in resourcelimited networks. 2.3.2 Link Layer Approaches The existing solutions in link layer can be divided into two categories, channel surfing and modification of MAC protocols for better network resilience against jamming attacks. There have been several papers [7, 35, 37, 47, 78, 83, 85, 87] regarding the investigation of jamming attack from the viewpoint of link layers. In this section, we describe various types of evasion techniques in the link layer. Even though the channel surfing method in the MAC layer was motivated by FH, it is the most reactive approach because it switches channels on demand after verification of jammed communications. Most channel switching schemes are reactive, which means that the shifting takes place only after communication is jammed. The reactive scheme is efficient, but channel monitoring hardware is needed. In the proactive schemes, most solutions are in physical layer and do not require a monitoring system, but result in low spectral efficiency in a resourcelimited network, because the frequency periodically has to hop into safe one. The critical issues for this solution are synchronization, latency, and scalability from the coordinated channel switching procedures across whole networks and the overhead from scanning all the channels. [83] introduced the channel surfing strategy, such that when the nodes with detection sensors are jammed, they switch their communication channel into another orthogonal channel in order to reconnect to the rest of a network. The boundary nodes that lose their neighbors from a jamming attack can discover lost neighbors in new channels and try to rebuild the connectivity of the entire network. There are two reasons nodes lose their neighbors, poor connectivity and jamming attacks, and they provide simple protocol to identify the reason for the lost neighbors by analyzing the channel being used for reconstruction by the lost neighbors. However, this protocol forces networks into an unstable state during connectivity rebuilding due to frequent link quality degradations or the dynamic behavior of networks. Two methods have been proposed in order to restore network connectivity after attacks. The first is the coordinated channel switching technique when an entire network switches its current channel to a new channel so as to reconstruct network connectivity. This technique suffers from unreliable links, so that some nodes might miss the notification to shift their channel to a new one. The second approach is the spectral multiplexing technique where boundary nodes act as bridges to connect the nodes of old channel to the nodes of new channel. This approach enables the networks to maintain multiple channels, so that the entire network does not need to notify all nodes, just some, to switch to a new channel. There are, however, several challenging problems to realizing a practical system, synchronization among the nodes with different channels, initiation of channel shifting from jamming attacks, and slot duration in a synchronous spectral multiplexing algorithm. [87] also described a spatial retrieval method, called a physical evasion method, by physical repositioning of mobile nodes out of jammed regions, but the networks would be unbalanced and even isolated by the attacks. Since they assume stationary jammers with mobile nodes, the nodes within a jammed region would be able to escape from the jammed region after the presence of jammers was detected. The challenging issues of the physical evasion method are how to determine which directions nodes should retreat and how far they should retreat from their current positions, because these decisions may cause disruption of network connectivity. In addition, this paper mentioned jammed nodes moving into radio range after relocations, but this would result in shorter network life time as well. As another type of evasion technique, [85] designed a timing channel to recover reliable communication links after jamming. The timing channel is a lowrate layer over physical/linklayers used to detect the timing of interfered packets in the receivers by utilizing CRC check or monitoring signal strength. This approach is for pointtopoint communication links, not for broadcast communications. The critical dependencies of the timing channel scheme are how to detect the exact timing of the failure packet receptions and how to map the occurrence of failed packets to the information to be delivered. [7] proposed a jammingresistant MAC protocol that adjusts the probability for successful transmission by monitoring channel activities, and each node would be able to transmit messages based on probability. The protocol also divides the time into smaller or bigger time intervals according to successful message transmissions in order to adapt transmission time. The main idea behind this is that adversaries observe the activity of the current channel and, if there is not enough activity, they would not heavily jam the channel. However, this protocol should include a mechanism to determine successful transmissions of messages in remote nodes, which means that other neighbors should be able to identify the result of transmission but communicators, and it is not only difficult, but is also burden to WSNs. In addition, if the sum of probability of each node is too high, then the nodes have less chance to observe the idle channel or a successful transmission of a message. A modified MAC layer protocol for defeating stealthy jammers based on IEEE 802.15.4based hardware was proposed by Wood [78] in order to reduce the damage from jamming attacks on communication packets. They introduced several strategies to defend the MAC layer according to the type of attack. For example, frame masking against an interrupting jamming attack, channel hopping against an activity jamming attack, packet fragments against a scan jamming attack, and a redundant encoding method against a pulse jamming attack. Frame masking is a DSSS technique using shared keys between wireless nodes. The packet fragmentation method would be used to transmit a message in multiple fragments during a jammer's channel activity scanning, and the redundant encoding method is useful for the receiver to recover corrupted messages due to jamming attacks. They want to combine all these techniques into a MAC layer protocol to defeat jamming attacks. The fundamental limitations on each defense mechanism are remained. That is, frame masking has a problem of key distribution, and channel surfing has serious synchronization problem in practical system as describe before. Packet fragmentation method might divide a packet into too small fragments with additional redundant encoding data for recovery, which makes the approach unfeasible in real communication systems. There have been two main approaches to jamming attacks in the link layer, channel surfing and modification of the MAC protocol. Both belong to a category called evasion methods, and utilize the jammers' scanning time to transmit legitimate messages. Channel surfing schemes from [83, 85, 87] are reactive in terms of switching channels, but synchronization is a critical issue to implement in a practical system. Modification of MAC protocol schemes require additional communication overhead among the nodes in the jammed area, which might be unfeasible under heavy jamming signal from dense networks. Link layer approaches are useful methods to detect jamming since the jamming attacks usually disobey the MAC layer protocol, but this requires additional study. 2.3.3 Network Layer Approaches There have been a few approaches addressing jamming attacks in the network layer, and those are related to scheduling of messages transmissions, jammed area mapping, or the linear programming model. Most of the defenses are focused on the physical and link layers, so network layer approaches are newly designed and still being investigated. A mappingbased evasion protocol has been introduced by [77]. In this system, the jammed nodes cooperatively map a jammed region. Jammed nodes that are within a jammed area transmit multiple blind messages to announce their jammed status to the mapping nodes that are not in the jammed area, but have jammed neighbors. The mapping nodes communicate with other mapping nodes to isolate the jammed area and to identify bridge nodes. The bridge nodes participate in relaying messages around the jammed area. One deficiency to this approach would be the possibly unnecessarily large jammed region built against the reactive jamming attack. As a result, parts of the network might be isolated. This is because many nodes in the exaggeratedly large jammed region may still be able to transmit without activating the jammers, yet they are isolated and the message deliveries are interrupted. During the mapping procedure among the mapping nodes, the protocol requires an excessive number of communication messages to build a detour route around the jammed region. The multiple traffic topologies from [62] could be used to evade the jammed nodes under attack from mobile jammers. The mobile jammers in this paper would be able to identify the critical broadcasting paths in order to prevent downlink nodes from receiving any messages. The nodes cooperatively construct the multiple paths and select a path based on the position of the jammers but, according to the paper, each sensor node needs to carry some secret and overall routing information before deployment into the network. This prior network information might not be feasible in dynamic networks and, due to the locality of sensor nodes, routing information in each node might not be consistent. In addition, the synchronization problem has to be considered in this protocol to change the current network topology into a new one. [63] designed a linear programming model for a specific type of the jamming attack, but it focuses mainly on a flowbased attack without considering of a protocolbased attack model. Unfortunately, this might not apply to general jamming attacks. [16] investigate an efficient scheduling technique for broadcasting messages when under a jamming attack. This approach shows good performance only when there are power limitation on jammers, which might not be a practical assumption since usually the jammers are much stronger than the normal nodes in WSNs. 2.4 Conclusion Through this investigation of various types of jamming attacks and existing solutions, we can conclude that, in reactive jamming attacks, the jammers stay idle until being triggered by messages disseminated within their transmission ranges, thereby further reducing the jammers' operation overhead and making it difficult to detect, thus this intelligent attack can be utilized by malicious users in more realworld scenarios. An efficient defense mechanism against a reactive jamming attack will be presented through this dissertation. CHAPTER 3 CENTRALIZED IDENTIFICATION OF TRIGGER NODES This chapter will focus on the identification of trigger nodes in a centralized manner, which can provide a general framework to build an efficient countermeasure for reactive jamming attacks in multichannels WSNs. By utilizing traditional group testing (GT) theory [22, 23] coupling with minimum collection of disjoint disk cover based grouping, this solution can identify all the trigger nodes with low overhead in terms of running time, computation and message complexity. The theoretical analysis and experimental results show that our solution performs well in terms of time and message complexities, which provides a good approach to defend reactive jamming attacks. 3.1 Network Model and Problem Definition The WSN in our problem consists of N sensor nodes, each having the same transmission range r and one base station (BS) with the transmission range p = pr where 3 > 1. Up to J < N static jammer nodes, whose transmission ranges are uniformly R = ar where a > 1, are deployed within the network. However, their positions cannot be known beforehand, except that all jammers are assumed to be sparsely deployed so that they can jam as large area as possible. Each sensor node or jammer node is equipped with k channels and m radios (m < k). We model the considered network as a connected graph G(V, E) where V is a set of N nodes and E = {(u, v)S6(u, v) < r, u, v e V} representing communication links between nodes. Any sensor nodes whose broadcasting can trigger some jammers are called trigger nodes, while any sensor nodes whose communications are interfered by jammers are called victim nodes. Therefore, any node v is a victim node if 6(J, v) < R for some jammer node J, whilst w is a trigger node if 6(J, w) < r. Note that trigger nodes are also victim nodes as the noise range (transmission range of jammers) R is larger than sensor transmission range. More importantly, since reactive jammers would keep track of the frequency shifting sequences or channel selecting mechanisms, excessive exposure of these methods against reactive jammers might be vulnerable to achieve effective communication performances among victim nodes. Since each sensor node has the same transmission range r and only the neighbor nodes within r can receive its message, the graph G(V, E) is a Unit Disk Graph (UDG). The objective of the problem is to find out all the trigger nodes within minimum time and message complexity. After the identification, a new routing path would be constructed to avoid activating any reactive jammers. Some notations used throughout this chapter are depicted in Table 31. Table 31. Notations Symbol Meaning r The transmission range of each sensor R The noise range of the jammers p The transmission range of the base station V The set of nodes in WSN N The number of nodes in WSN W The set of victim nodes in WSN W, The set of left victim nodes in WSN after cover i n The number of victim nodes in WSN .N, The number of victim nodes covered in cover i ni The number of victim nodes before cover i nj The number of victim nodes in group in cover i U The set of trigger nodes in WSN d The number of trigger nodes d, The number of trigger nodes in group in cover i k The number of channels in WSN m The number of radios in WSN A(G) The maximum node degree of graph G ,(D) The number of nodes disk D covers 6(u, v) The distance between two nodes u and v H(6) Unit Disk Graph H with disk radius 6 ti The total number of testing rounds in cover i C The total number of testing covers T The total testing round 3.2 Preliminaries In this section, we introduce some preliminaries on MAXIMUM CLIQUE PROBLEM and NONADAPTIVE GROUP TESTING, based on which we discuss how to apply them to our problem. 3.2.1 Maximum Clique Problem The Maximum Clique Problem is defined as follows. Given an arbitrary undirected graph G(V, E), a subgraph G'(V', E') (V' e V) is a clique if all its vertices v' V' are pairwise adjacent. The maximum clique is a clique with max I V'. The maximum clique problem is also one of the first problems shown to be NPcomplete [9]. So far, the best polynomialtime approximation algorithm for the maximum clique problem was developed by Boppana and Halldorsson [9], and achieved an approximation ratio of n('1o()). In [9], Hastad shows that this is actually the best we can achieve and it cannot be approximated within a factor that of n1' for any C > 0. There are some other results in the literature concerning the approximation of the maximum clique problem on arbitrary or special graphs [9, 10, 29]. In this chapter, the maximum clique problem is applied to obtain the upper bound of the number of trigger nodes based on the number of reactive jammers. Since a jammer can only be activated by the nodes within a certain distance, we can construct a unit disk graph of all nodes with the radius twice the distance to estimate the upper bound of the number of trigger nodes. 3.2.2 NonAdaptive Group Testing Nonadaptive Group Testing (GT) [22, 23] methods are to minimize the testing period by sophisticatedly grouping and testing the items in pools simultaneously, instead of individually testing them. The way of grouping is based on a 01 matrix Mt,, where the matrix rows represent the testing group and the each column refers an item. M[i,j] = 1 implies that the jth item participates in the ith testing group, and the number of testing is the number of rows. The result of each group is represented as an outcome vector with size t where 0 is a negative testing result (no trigger in this testing group) and 1 is a positive result (some triggers in this testing group). To achieve the minimum testing length for nonadaptive GT, M is required to be ddisjunct [23], where the union of any d columns does not contain any other column. Based on the properties of ddisjunctness, the decoding algorithm to identify the triggers based on the testing results becomes very simple. We just need to remove all the items appeared in any negative pools and the remaining item are positive [23]. In this way, only 0(1) testing rounds and O(tn) decoding time are needed. To utilize GT for our trigger detection, we need to solve the two most challenging problems: (1) How to group the nodes to avoid interference between the results among groups so as to test these groups simultaneously. (Any two groups are called interference free if any jammers triggered by either group cannot jam the other group). (2) How to accurately estimate the value of d which is the upper bound of the number of trigger nodes. Since d determines the number of tests, the tighter d is, the better time and message complexities we can obtain. 3.3 Centralized Trigger Node Identification (CTNI) In this section, we devise CTNI algorithm to identify all trigger nodes in WSNs so that reactive jamming can be avoided when these trigger nodes do not transmit messages. The basic idea of CTNI is as follows: We first detect all victim nodes WSNs by using any existing scheme such as existing alarm forwarding scheme [60]. Then we test these victims to identify the trigger nodes by calling two subprocedures: 1) We use the GVNMCDDC algorithm Group Victim Nodes Based on Minimum Collection of Disjoint Disk Cover to group as many as victim nodes without interference with each other in each cover. Each cover includes a set of disjoint disks where the center of each disk will act as a test outcome collector. Each of the disjoint disks can be tested simultaneously. 2) For a set of victims in each disjoint disk, we use the DTNNCGT algorithm Detection of Trigger Nodes based on Nonadaptive Combinatorial GT to detect all trigger nodes within these victim nodes. During the group testing process, the collectors will collect the test outcome and perform the decoding process. We continue covering and testing victim nodes until all victim nodes are tested. The details of two major algorithms are elaborated next. 3.3.1 Group Victim Nodes Based on Minimum Collection of Disjoint Disk Covers (GVNMCDDC) Algorithm After identifying all victim nodes, the goal of GVNMCDDC algorithm is to group victim nodes so as to simultaneously test as many interferencefree victim nodes as possible. The basic idea of this algorithm is as follows. For each victim node v, construct two disks Dv and D2 centered at v with radius (R r) and (3R r) respectively. The objective is to obtain a minimum collection of disjoint disk covers, where each cover is a set of disjoint disks such that victim nodes within each disjoint disk can be tested simultaneously without mutual interference. (Any two groups are called interference free nodes if any jammers triggered by either group cannot jam the other group.) We adopt a greedy method for this by selecting a node v whose corresponding R r disk covers the maximal number of victim nodes, and then select another node u similarly after removing all the nodes within 3R r from v from the graph. Iterate this until no node left in the graph. 3.3.2 Detection of Trigger Nodes Based on NonAdaptive Combinatorial Group Testing (DTNNCGT) Algorithm Now, after finding the collection of disjoint disk covers, we start conducting the group testing for each cover. Notice that for each cover which consists of a set of disjoint disks, we will conduct the test for each disk simultaneously. For each diskj, i.e., Dv, in cover i, gather all the victims in this disk into a group Gy for testing based on the nonadaptive group testing technique. In order to apply this technique, we need to estimate an upper bound Dy of the number of trigger nodes in each disk as shown from line 7 to 11 in Algorithm 2. Based on this obtained value Dy, DTNNCGTconstructs Algorithm 1 The GVNMCDDC Algorithm 1: Input: All left victim nodes W,_I after cover / 1 2: Output: The collection of groups in all covers G,1 ..., G 1 < i < C, 1 4: while I WI / 0 do 5: > Construct double disks for each victim node 6: for w e W,_ do 7: Construct D' and D 8: end for 9: k 1 10: ,W < 0 11: while I W,l 0 do 12: Choose w e W;_, to maximize K(Dl) 13: Gik D' 14: W,_, W,_ \ D2 15: 2 15: W,  W, U { D2 \ D } 16: k k + 1 17: end while 18: W  W 19: end while a D,disjunct matrix accordingly. From this matrix, m pools (rows) will be tested at the same time. The nodes in different rows broadcast the test messages in an orthogonal way (i.e. on different channels) so that the testing result will not interfere with each other. Consider the nodes in one group broadcast on the same channel, the jammer is supposed to be activated and broadcast noise on this channel if some nodes in this group are trigger nodes. Then the collectors will collect all the testing results in this disk and start the decoding procedure. Or else, they can transmit the results to base stations and the decoding procedure will be performed at the base station. During communication with the base station, the collectors are required to perform channel surfing method for successful delivery of the outcomes. By decoding these results, trigger nodes will be identified. Finally, the complete CTNI algorithm is presented in Algorithm 3. Consider Figure 31 as an example where we have two jammer nodes J1 and J2. Nodes vl, v2,..., v9 and v15, v16,..., v25 are the victims, and m = 3. According to our Algorithm 2 The DTNNCGT Algorithm on Group j in cover / 1: Input: Victim nodes set W, in one group, R, r 2: Output: Number of trigger nodes Dy in this group 3: Construct Gy = (Wy, E), where E = {(u, v)16(u, v) < 2r, u, v c WVy} 4: 5: > Find the upper bound Dy 6: Dy  0 7: for k = 1 to Jy do 8: Find the MAXIMUM CLIQUE c(Gy) on graph Gy 9: Gy  Go \ Uwc(G,) w 10: Dy Dy + Ic(Gy) 11: end for 12: > Test by using NONADAPTIVE GT 13: Construct a DyDISJUNCT MATRIX M 14: Group the column in each row with entity 1 into one group 15: Test these groups simultaneously 16: Decode the testing result to identify all trigger nodes Algorithm 3 The CTNI Algorithm 1: Input: WSN G(V, E) 2: Output: TNLT Broadcast Tree T 3: W  The set of victim nodes 4: W,  The set of left victim nodes from cover / 1 5: Gy < The group j in cover i 6: U  The set of trigger nodes 7: U,  The set of trigger detected in cover i 8: T  The TNLT broadcast tree 9: W 0, U  0, W  Victim nodes 10: W,  W 11: = 1 12: while W,I > 0 do 13: Gy  Groups based on the GVNMCDDC algorithm in cover i 14: Ui j trigger nodes based on the DTNNCGT algorithm 15: UUU U, 16: i  i+ 17: end while Figure 31. Since item 6 (6th column) is a trigger node (positive item), only the 2nd and 6th groups (rows) return negative outcomes. On the contrary, all other four groups produce positive outcomes. algorithm, two disjoint disks will be found and two groups G1 {v1, v2,... v} and G12 = {v15, v16 ..., v25} are constructed accordingly. Testing will be conducted on these two groups simultaneously. For simplification, Figure 31 just shows the detail testing of Gil. After the estimation of Dy = 1, our algorithm will construct a 1disjunct matrix. Based on this matrix, the first three rows will do a onehop broadcast message to three 44 channels accordingly (since we have m = 3 in this example). More specifically, nodes v1, v2, v3 send a test message on channel 1, nodes v4, v5, v6 send a test message on channel 2, and v7, v8, v9 send a message on channel 3. v, is a center of this disk and will act as a collector. As v6 is a trigger, then the test outcome on this row (second row) is positive. Next, the last three rows will broadcast the test message. After finishing all rows, v1 will have a outcome vector as shown in Figure 31 where the second and sixth rows have a positive result. Based on a simple decoding method mentioned earlier, we can easily detect v6 as a trigger node. 3.4 Theoretical Analysis 3.4.1 Estimation of Trigger Node Upper Bound Dy In order to construct ddisjunct matrix for testing in testing group Gy, we need to obtain an upper bound on trigger nodes. We assume that the interference radius is larger than legitimate transmission radius, R = ar where a > 1 since jammers have more capabilities than normal sensor nodes. Let J be the set of jammers that trigger node t could activate. We note that the distances from jammers in J to t are at most r while the distance between any two jammers must be larger than R = ar. Otherwise jammers will invoke each other and run out of energy. We have the following lemma: Lemma 1. Let J be the set of jammers that the trigger t could activate, then J1 < 7T arcsin(') Proof. Let 0 be the location of t. Assume that J contains jammers with locations Ji, J2,..., Jm in clockwise order like in Fig. 32, where m = J. We have OJ,(= 6(0, Ji)) < r Vi = 1... m and JiJj > R = ar VI < i < j < m. m Since JOJ,+i 2=7 where Jmr = J1, let J OJi = 3 be the smallest angle, we i=1 have j < 2 and J < 2. Use the cosine's law: R2 < JJ = OJf + OJ2 20JOJ+l cos/3 As jammers will not revoke each other, we have JJ,+l > R > r > max{OJ,, OJi,+}. Hence, ZJOJ,+ will be the largest angle of the triangle JIOJ,+ We obtain 3 > 2 i.e. m < 6. From 3 > 2, OJ2 + Oj2+ 20JOJ,+; cosp obtain the maximum value at OJ, = OJ,+ = r. Hence, a r2 < r2(2 2 cos ) or / > arccos(1 ) = 2arcsin(2). 2T Therefor, J = m < 2n( 2arcsin(') Following the lemma, we have: * IJ < 1 when a > 2. * J < 2 when a > V3. * J< 3 when a > V2. * J < 4 when a > 5 5/ * J < 5 when a > 1. *Jammer *Trigger Node Figure 32. 5 Possible jammers activated by a trigger node t Theorem 3.1. The upper bound Dy of the number of trigger nodes in one group is UCk((Gy) k= 1 where ck(G) is the kth maximum clique on graph G. Proof. According to LEMMA 1, the nodes in group Gy can trigger at most Ju jammers. Intuitively, we know that a set of trigger nodes to activate the same jammer have a distance less than 2r. In algorithm 2, we construct a unit disk graph G, = (Wy, Ed) with disk radius 2r so that the nodes which trigger the same jammer must form a clique in graph Gy. In each iteration, according to Algorithm 2, we choose the first jth maximum cliques and union all these cliques. That is, uJ l ck(Gu) Thus the proof is complete. 3.4.2 Correctness Lemma 2. Any two nodes with the distance larger than R + r are interferencefree nodes. Proof. Assume that any two nodes u and v with distance 6(u, v) > R + r are not interferencefree. Then there exists a jammer J such that J can interfere both u and v. Without lost of generality, we can assume that node v activates J. Thus 6(v, J) < r. Plus, 6(v, J) < R and 6(u, J) < R, then 6(u, v) < R r, which contradicts our assumption. O Lemma 3. For each set of victim nodes in disk Dv, the center node v can be used as the collector. That is v can sense the noises from any jammers triggered by any nodes within the distance R r from v. Proof. The proof is straightforward. Assume that a center node v in disk Dv cannot sense the noise from a jammer J, which is activated by a node u in the disk Dv. Then, we have 6(u, J) < r and 6(v, J) > R. Therefore, 6(u, v) > R r, contradicting to the fact that u is in D'. O Theorem 3.2. The CTNI algorithm can correctly identity all trigger nodes. Proof. Since the jammer noise range R is always larger than normal transmission range r, the trigger nodes must be included in the victim nodes. Therefore, if we test all victim nodes, we must be able to identify all trigger nodes. Note that from Lemma 2 and the fact that each disk Dv has a radius (R r), all the victim nodes in any two different disjoint disks are interferencefree. Thus the testing result is correctly collected. D 3.4.3 Performance Analysis Lemma 4. Given the UDG H with radius 3R r, denote the maximum node degree in H as A(H), the total number of rounds (where in each round, we use several disjoint disks D2, D2, . centered at node u, v, .. to cover nodes, i.e., contain as many nodes as possible in their corresponding concentric circles D,, D,, ) needed to cover all the sensor nodes is at most A(H). Proof. This is a loose upperbound and can be obtained by considering the disks in the same round as an MIS (Maximal Independent Set). For the ith round, denote the maximum node degree of the current graph H as A,, and the set of any center u of the selected disks D,2 form an MIS of H, then the size of such an MIS is lower bounded by W1, where I W,I refers to the number of uncovered nodes at the beginning of this round. Henceforth, the number of nodes covered in the ith round at least equals to the size of this MIS, i.e. Wl" Ai 1" Since the number of uncovered nodes is decreasing round by round, A, is nonincreasing for each round, so straightforwardly at most A(H) + 1 rounds where A(H) = A1 = maxi A, are needed. O Lemma 5. The number of testing covers to detect trigger nodes in each group of victim nodes no is upper bounded by D' log n0 [min {(2 + o(1)) Iog2 (0 o n,}/m log (Dy log, ny) where Do = U Ck 1C(G) Proof. The best upperbound of the number of rows for ddisjunct matrix is min{(2 + o(1)) jo ) n}, using Du's construction [22, 23]. In WSNs, as we defined there are m radios so that at most m groups can be tested at the same time. According to THEOREM 3.1, do are bounded by Dy and n. is the number of victim nodes, we complete the proof. D Corollary 1. The total number of testing rounds in cover C, is upper bounded by D log n max[min {(2 o(1)) D02 u nn,}/m] j log2(DU logn2 ) Theorem 3.3. The total testing round T is upper bounded by A(H)+1 02 no2 maxFmin {(2 o(l)) '2 n}/m] SIog(D log12 no)' where D = Uk Ck c(G) Proof. According to LEMMA 5 and COROLLARY 1, the covers for all victim nodes are A(H) + 1 and the testing time for each cover is the maximum testing time among all groups, that is, D log nd maxFmin {(2 o(1)) DJ2 n' n,}/m] j log2(DU logn2 ) where D. = UJ 1 ck(G) The proof is complete. D Theorem 3.4. The Message Complexity per node w is (2 + o(1)) 0DOnn)  Proof. In D,disjunct matrix, the number of messages each node needs to transmit is the number of 1entries in the corresponding column. As we mentioned above, Du's construction method [22, 23] for ddisjunct matrix, has the lowest upperbound for the matrix size. It is trivial to find that, each column has exactly s 1entries in the matrix constructed in that way, where (D, log, n, s = (2 + o(l)) D092n log2(D, log2 n,) hence the message complexity per node is the same. O We do not consider the false negative from random delay on emitting adversarial signal from jammers since according to the definition of reactive behavior, the jammers would only emit interference signals during the legitimate activities on channels. 3.4.4 Random Reactive Jamming Model Our proposed solutions and analysis are based on a simple reactive jamming model, of which a jammer will start jamming right away whenever it senses some transmission activities. However, jammers may adopt a more sophisticated model to evade the detection by not responding to some messages. This makes identification approach more challenging due to the inaccurate testing outcomes. Fortunately, the identification concept proposed in this work is still applicable. Under this attack scenario, instead of using ddisjunct matrix during the testing procedure, we will adopt the (d, e)disjunct matrix, which is a binary matrix so that for any column c, there exist at least e + 1 rows where c has a 1entry but none of other d columns has. There are several studies and results on the construction of such a (d, e)disjunct matrix that we can apply [22, 23]. Using (d, e)disjunct matrix helps to correct at most e errors in the testing outcomes, thus we are still able to correctly identify all the triggers. We would like to note that in order to use the mentioned (d, e)disjunct matrix, we need to estimate the upper bound of e. In practical network environments under random reactive jamming attacks, we could estimate this bound by analyzing the Packet Delivery Ratio (PDR). PDR is a ratio of the number of packets which are successfully validated through the Cyclic Redundancy Check (CRC) procedure and defined as: PacketsPassedCRC PDR = ReceivedPackets The reactive jammers would be able to drop PDR effectively by the reactive strategy, however, we could use this ratio against them to bound the unreliable testing outcomes in the case of random reactive jamming behaviors because Pr(1 PDR) is the probability of emitting adversarial signal from jammers, and PDR is also one of wellknown probabilistic methods to determine the presence of jamming attacks by a simple calculation. In order to achieve highly accurate PDR, Strasser et al. [60] introduced a biterror identification technique to differentiate jammed packets from errors caused by weak signal (e.g., because of fast fading or shadowing). Consequently, the upperbound of errors over tests could be derived from investigation of PDR and sent from sensors to base station so as to construct errortolerant disjunct matrix. 3.5 The TNLTCDS Routing Algorithm One of the benefits for identifying the trigger nodes is to help construct a routing protocol which does not activate any reactive jammer. In this section, we discuss a simple routing algorithm called Trigger Nodes Leaves Tree based on Connected Dominating Set (TNLTCDS) which uses trigger nodes as only end receivers. Together with the CTNI algorithm, TNLTCDS will complete an efficient countermeasure for reactive jamming attacks. We utilize the Connected Dominating Set (CDS) to construct our TNLTCDS as CDS has been shown as one of the most efficient methods for constructing a broadcast protocol. Again, consider network G = (V, E) with U c V as a set of trigger nodes identified by CTNI. We will construct a directed graph G' = (V, E') by changing all the undirected edges (u, v) c E where u e V \ U and v c U to the directed edge (u, v). We then deploy a CDS algorithm in directed graph [64] on G'. It is easy to see that the obtained CDS S will not consist of any node in U. Finally, we construct a broadcast tree T by connecting nodes in S to the rest using newly added directed edges. This simple idea is to show that it is quite easy to implement a routing algorithm where triggers are as only receivers. Thus, the approach of trigger identification is a very useful concept in the defense of reactive jamming attacks. 3.6 Performance Evaluation In this section, we evaluate the efficiency of our design through a series of simulations in terms of time latency and message complexity for sensor networks with different parameters. The results of these experiments show that the proposed solution is timely efficient for identifying trigger nodes and defending reactive jamming attacks. S 400 350 : : / 300 250 S \ 200 max node degree A 150 / # of total rounds T # of disk cover c max # of testing rounds per disk t 1 2 3 4 5 6 7 8 9 10 11 0 #ofjammersJ 1 2 3 4 5 6 ##5amefsJ # of messages M # of victim nodes d 7 8 9 10 Figure 33. Experimental results by various size of jammers 3.6.1 Simulation Setup In order to simulate a general sensor network, we randomly distribute a total of N sensor nodes with one base station and J jammers to a square network field with width s. As has been mentioned above, the base station, sensor nodes and jammers have respectively transmission range, p, r and R. In order not to exaggerate the power of the B 200 300 180 160 1 / \ \ 250 , 140 120 200 100 80 max node degree A 150 6 # of total rounds T # of messages M 60 # of disk covers c 40 max # of testing rounds per disk t 1 # of victim nodes d 20 \  100   0 1 2 3 4 5 6 7 8 9 10 11 50 2 3 4 5 6 7 8 9 10 010 # of channels m # of channels m A B Figure 34. Experimental results by various size of channels 200 240 180 220 160. 200 140 1/ / 180 120 160 100 140 80 max node degree A # of messages M 60 # of total rounds T 12 # of victim nodes d 60 # of disk cover c 100 40 max # of testing rounds per disk t 20 060 500 550 600 650 700 750 800 850 900 950 1000 40' 500 600 700 800 900 1000 # of nodes N # of nodes N A B Figure 35. Experimental results by various size of nodes base station, we assume p = r in this simulation, while larger p would make this solution more efficient. We have in total six benchmarks in the simulations with different input parameter teams. On one hand, we study the average number of disk covers c in the GVNMCDDC algorithm, and the maximum node degree A to validate the bound of c proved in LEMMA 5. On the other hand, we show the overall test length (number of rounds T) analyzed in THEOREM 3. Moreover, we record the number of victim nodes n and the total volume 150 140 250, 130 max node degree A 120 # of total rounds T 110 # of disk cover c 100 max # of testing rounds per disk t 200 100 90 80 70 7 150 60 50 / 40 30 100 20 # of messages M 1 0 00 # of v victim nod es d 10 2 0 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 50 width of square network region s (*103) 1500 2000 2500 width of square network region s A B Figure 36. Experimental results by various network densities 800 8 U .700 / 00 600 600 600 500 500 400 400 300 max node degree A 300 # of total rounds T # of messages M 200 # of disk cover c 200 # of victim nodes d max # of testing rounds per disk t 100 2 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 020 22 24 26 28 noise range ratio a noise range ratio c A B Figure 37. Experimental results by various size of a of communication messages M between the sensors and the base station, to indicate the message complexity of this method. To investigate the effects of a series of network parameters, over the efficiency of this solution, we vary the values for the number of jammers J, number of radios m, number of sensor nodes N, width of the square net work region s as well as noise range ratio a, hence the following five paragraphs and Figure 5.2.1, 3.6, 3.6, 3.6 and 3.6 are the corresponding results and analysis. Note that for each parameter team, 100 network instances are investigated and the results are averaged. 3.6.2 Results and Analysis 3.6.2.1 Performance by the number of jammers J Figure 5.2.1 (a) and (b) explain our protocol performance based on the various numbers of jammers J in the network. In this test, we have N = 1000 nodes with m = 3 radios, on a 1500 x 1500 network field, where J e [1, 10] jammers are randomly deployed. Our protocol employs a sophisticated technique to perform as many parallel testing as possible as shown in Algorithm 3, therefore the number of testing rounds, T, can be stable while the number of jammers J and victim nodes n increase. As shown in Figure 5.2.1 (a) and (b), T increases a little while n can vary from 50 to 450 when J increases from 1 to 10. More specifically, the number of disk covers c and maximum number of testing rounds per disk t are smaller than 10, where the latter is much smaller than maximum node degree A. This contributes to dramatically small number of overall rounds T, which is no larger than 30 and stable for increasing J. Moreover, since each (R r)disk in our tests needs only one sensor node to send the result back to the base station, the message complexity M is also much smaller (less than 100) than the number of victim nodes n. Note that in individual testing method, M should be as high as 0(n). Therefore, our solution can promptly defend a jamming attack with increasing number of jammers, in terms of time complexity and message complexity. 3.6.2.2 Performance by the number of radios m During the series of tests within (R r)disk, we accelerate the overall testing latencies by employing the multiple radios m since with a given ddisjunct matrix, m number of rows can be tested simultaneously. The parameters for this simulation are randomly distributed N=1000 nodes and J=5 jammers in a 1500 x 1500 network area, where m e [1, 10]. In other words, the number of tests within an (R r)disk can be reduced by the factor m, which also shortens the overall test length. As illustrated in Figure 3.6 (a), the maximum testing rounds per disk decrease as the radio size increase, which assists to drop the total rounds, T, drastically. Especially, when m=2 from m=1 in the Figure 3.6 (a), the overall total rounds drop rapidly. In conclusion, we learn that the radio size can highly benefit the overall test length of our protocol. 3.6.2.3 Performance by the number of nodes N The number of nodes in the network is one of the critical aspects to consider a mobile network solution in terms of scalability. For instance, the countermeasure using DSCDMA technique suffers from the number of codes for encoding/decoding messages in each node of the network, since the newly joined nodes trigger the creation of additional codes for the victorious battle against jammers who try to corrupt the messages by pilfering the codes. However, the fact that our solution is a diskbased test approach by exploiting parallel testing relieves the scalability problem fairly, and the performance shows somehow constant movement where N e [500, 1000] in Figure 3.6(a) and (b). As shown in Figure 3.6(a) and (b), the victim quantity increases obviously as the number of nodes increases, but the number of messages is quite constant. Moreover, the total testing rounds increase slowly. This figure shows how our system efficiently operates when the number of nodes increases from N=500 to N=1000 with m = 3 and J=5 jammers in a 1500 x 1500 network area. From this evaluation, we can conclude that our model is also a very suitable security solution for the majority of sensor networks in various areas. 3.6.2.4 Performance by the density of the network Now, we show how the protocol we proposed reacts in the various network densities where the network field size broadens. With the given number of jammers and the increase of the network field, it is clear that the number of victim nodes decreases where the system tries to deploy the nodes in order to cover the network field as much as possible. As we discussed before, due to the fact that our approach is diskbased classification of the nodes, sparse network would mainly help to reduce the number of victim nodes, especially A, and then reduce the overall testing rounds as well. Figure 3.6 (a) and (b) shows the various simulation results with the increasing network field size from 1500 x 1500 to 2500 x 2500 where N=1000 with m = 3 and J=5 jammers. As the network is sparse, the number of victim nodes decreases as A gets smaller in this figure as we discussed. 3.6.2.5 Performance by transmission range of the jammers Now let us consider the interference range of the jammers. Since noise range is relatively larger than transmission range of sensor nodes, more messages of the sensor nodes will be jammed, thereby increasing the number of victims nodes. However, the number of trigger nodes may be kept the same. It implies that JAM [77] locks down the whole jammed region while our system minimizes the jammed region size by identifying the smaller number of trigger nodes. In Figure 3.6(a) and (b), as the a gets larger, the number of victim nodes increases since a jammer can transmit farther and contaminates more nodes during the activation. Moreover, more victim nodes requires more testing rounds to cull out the trigger nodes among them. In our result, the number of rounds rises as a gets larger. However, the number of rounds is changing very slowly. 3.7 Conclusion To efficiently tackle reactive jamming attacks in multipleradio WSNs, we devise a new mitigation for identifying trigger nodes, whose broadcasting triggers the jammers, and a routing protocol to switch trigger nodes to receivers so as to keep jammers idle. By utilizing nonadaptive group testing scheme, disjoint disk cover method, and cliquebased clustering, this countermeasure achieves low overhead in terms of time and message complexity, thus is practical for general WSNs. Besides the analytical complexity analysis, we also conduct a series of simulations to investigate the scalability and stability of this method to various WSNs. Throughout this chapter, we assume no packetloss during all the transmissions, while in real WSNs this is inevitable. In the case that all the broadcasting messages of one trigger node do not arrive to the jammers nearby, due to the packetloss, the test outcomes of the corresponding groups might have error and fail to identify all the trigger nodes out. Even in this extreme case, our method can also be adapted using errortolerant GT techniques [22, 23] discussed in Section 3.4.4. CHAPTER 4 LOCALIZED IDENTIFICATION OF TRIGGER NODES The centralized approach from previous chapter cannot avoid inevitable burden of the excessive communication overhead to base station, which can not realize a prompt recovery system under the severe jamming attack. To this end, this chapter designs a localized trigger detection scheme which employs two techniques: a hexagon tiling coloring scheme and a sequential group testing (GT), which efficiently identifies all trigger nodes without requiring any extra hardware on sensors. Different from our previous work, this method can be distributedly implemented with a low time and message complexity. Furthermore, a more sophisticated behavior of reactive jammers is also investigated where they do not start injecting noise as soon as they sense some signals. Instead, jammers randomly start jamming with some probability p under this new model. 4.1 Network Model and Problem Definition We consider WSNs consisting of n sensors having a uniform transmission radius r. Thus the WSN can be abstracted as a Unit Disk Graph (UDG) G = (V, E), where V represents the set of n sensors, and E = {(u, v) d(u, v) < r, u, v e V} representing communication links between nodes Up to J < n static reactive jammers at unknown locations, whose transmission ranges of interference signals are uniformly R = ar where a > 1, are deployed within networks. Since the basic goal of reactive jamming is to disrupt legitimate packets with minimum energy cost and maximum stealthy, the damages from the reactive jammer are limited to specific sensors on specific transmission channels by short adversarial signals. Due to the fact that most of wireless sensor devices terminate transmissions as soon as detection of jamming signal through existing detection methods, it is unnecessary for the reactive jammers to drag down its energy efficiency by transmitting interference signals for a long time. Moreover, from the standpoint of the jammer, d(ji,j2) > R should be satisfied for any two jammers ji and j2 J in order to avoid mutual invocation to each others Random Reactive Jamming Model: We relax the jamming model and consider that after sensing some activities, a jammer randomly starts jamming with probability p Any sensor node u e V is said to be a trigger node if there is a jammerj e J, such that d(u,j) < r. and a sensor node v is said to be a victim node, iff d(v,j) < R. The objective of the problem is to find out all triggers within minimum time and message complexity in order not to limit the actual jammed area by transmission range of R but r. 4.2 Overview and Fundamental Results In this section, we briefly present an overview of our localized solution along with some fundamental results that will be used later. Our localized algorithm can be implemented as a network maintenance service and can be periodically invoked to identifying all the trigger nodes. We will use testing to identify these trigger nodes by allowing them to send out a test message and listen if their is any noise. Especially, [60] proposed a highly accurate detection scheme of reactive jamming attack, and all interference signals can be correctly identified as noise by sensors from other external interference even in lower and unsteady RSS. However, individual testing is too timeconsuming, thus we often test a number of nodes, called group testing (GT). However, testing a group of nodes simultaneously encounter several difficulties. For example, if some Noise sensed after performing testing, we may not know which ones in the tested nodes triggered the jammers. Moreover, scheduling nodes in a testing group to perform the testing synchronously may result in a lot of communication overhead in the network if tested nodes are far from each other. In addition, if two group of nodes are testing at the same time and the jammer triggered by the first group can jam nodes in the second group, the testing result may be inferred incorrectly. Hence, an efficient grouping and scheduling mechanism is essential to reduce the overall latency. We utilize two principles to efficiently reduce the number of testing rounds i.e. overall latency: 1. If two nodes u, v are at the distance at least R + r they cannot trigger a same jammer. This enable us to test u, v in a same round without having the outcome of testing u and that of v interfered each other. In general, we can perform testing in parallel for two sets of nodes U and V that are R + r far away from each other. 2. If u, v, w are identified triggers, then all nodes inside the triangle whose vertices are u, v, w are also triggers. Furthermore, if T = {t1 .... tk} is a set of triggers, then all the nodes inside the convex hull of T are also triggers. This holds as long as R > 2r. 4.2.1 Overview of Identification Procedure The overview of trigger identification procedure is depicted as follows: The sets of nodes are locally divided into hexagonal groups, and each hexagonal group is colored into disjoint interferencefree groups, where the transmission of nodes within a group will not activate the same jammer whose adversarial signals will disrupt the communications with other groups. The diameter of each hexagon is small enough so that all nodes in a same hexagon can communicate with each other directly and hence the latency of forwarding messages is avoided. These groups are called testing groups in remainder of this chapter. The set of nodes within testing groups with the same color are then scheduled to fulfill a sequential GT procedure simultaneously in order to identify all trigger nodes over each testing group. Notice that all nodes in a network do not need to exchange additional messages in order to partition themselves into disjoint groups (to be explained in detail later) and run a sequential GT procedure since the basic premise of our network systems are loosely synchronized in the order of seconds. The trigger identification algorithm within each pool use Sequential Group Testing (SGT) to identify only nodes that belong to the convex hull of the set of triggers that is often much less than the number of triggers. In principle, the flow of the identification procedure is: (1) Partition the set of nodes into hexagonal testing groups; (2) Assign colors to hexagons in order to maximize the number of disjoint interferencefree testing groups and schedule them according to the colors; (3) Perform sequential group testing within each hexagonal testing group during the assigned time slot for the each group, in order to discover all trigger nodes. How to divide the set of nodes into interferencefree testing groups and how to discover all trigger nodes within minimum latency play fatal role in our localized approach against reactive jamming attack. These will be illustrated along with theoretical analysis in the following sections. 4.2.2 Hexagon Tiling Coloring In this section, we propose a hexagon tiling coloring, which is the fundamental part that will be used later in section 4.3 to locally partition sensor nodes in a given WSN into a set of testing groups. In order to study the hexagon tiling coloring, we consider the following new problem: Definition 1. Hexagon tiling coloring problem: Given a distance d e R+ and a hexagon tiling H dividing the 2D plane into regular hexagons of sides '. Find the minimum number of colors needed to color H, such that any two hexagons h, and h2 in H with same color are at distance greater than d. The distance between two hexagons hi and h2, denoted as d(hl, h2), is defined as the Euclidean distance between any two closest points pi and P2, such that pi is located in hi and P2 is located in h2. This makes the hexagon tiling coloring problem different from the channel assignment problem [56] in cellular network, where the distance between two hexagon cells is measured from their centers. centers of all the hexagons are placed on a triangular lattice. Therefore, we consider a new coordinate system in the 2D plane, with axes inclined at 600. This new coordinates system has two units vectors 7 (, 0) and j (v, 3) as shown in Figure 41. The centers of each hexagon h coincide with the integral coordinates in this coordinate system. Now, each hexagon h can be identified by the coordinates (i,j) of its center as h(i,j). The Euclidean distance between two hexagon centers h(i1,jj) and h(i2,j2) is given as dc(hl, h) = /(i i2)2 +(i 2)(ji J2) + i J2)2 4.2.3 The k2Coloring Algorithm We now present our k2coloring algorithm for the hexagon tiling coloring problem. For a given distance d c R+, the k2coloring algorithm uses k2 colors, where k = 3 + 1 to color the entire hexagon tiling and guarantees that any two hexagons hi, h2 c H with d(hi, h2) < d have different colors. Figure 42 shows the coloring pattern generated by the k2coloring algorithm for d = 3 and k = 4. The k2coloring algorithm 2 is used by the sensor nodes in our proposed localized algorithm to locally identify the group they belong to. Algorithm 4 k2coloring algorithm Input: Given a hexagon tiling H and a distance d e R+ OutpuColored H Compute k = + 1] for all hexagon h(i,j) c H do Colorh(i,)  (j mod k)k + (i mod k) + 1 end for Lemma 6. For a given d c R+, the k2 coloring algorithm colors the hexagon tiling H, such that two hexagons hl, h2 c H have different colors if d(hl, h2) < d. Proof. The lemma can be proved by showing that in the color assignment generated by the k2coloring algorithm, for any two hexagons h((i,,jj), h2(i2,j2) e H, the distance d(hi, h2) will be greater than (k 1) (where k = [2 1). If h1(i, ji) and h2(i2, j2) are assigned the same color by the k2coloring algorithm, then1 (mod k) x k+ i (mod k) + 1 =J (mod k) x k+ i2 (mod k) + 1. This happens iff i /2 (mod k) ji j2 (mod k) (x=k, y=0) (\3/2) x (k1) ( 3/2) x (k1) (x=k, y=0) (3/2) x (kl) (,3/2) x (kl) (x=O, y=k) (x=k, y=k) Figure 41. The minimum distance between two nodes with same color Let x = il /2 and y = j j2. It follows that x and y will be multiple of k. The distance between the centers of hi(i,ji) and h2(i2,J2) is given by dc(h, h2) = x2 + xy y2 2 Consider the following cases: If x > 2k: The distance between the centers of hi and h2: dc(hi, h2) = ( + y)2 X2 > 6(2k)2 > (k 1) + 1. Note that for every hexagon the distance from a point inside it to its center is at most 1. Hence, the distance between two hexagons d(hl, h2) will be at least dc(h, h2) 2(1) > (k 1). If lyl > 2k: We also obtain the same result as in the case xl > 2k. If x and y < 2k: If x = y = k, then the distance between two hexagons d(hi, h2) will be at least dc(hl, h2) 2(1) > k2 1 > (k 1). Otherwise there are only six left cases of x, y as shown in Figure 41. The distance between two hexagons in all of these cases is exactly (k 1). Hence, the lemma is completely proved. F Figure 42. The coloring pattern for k = 4 4.3 Localized Trigger Node Identification (LTNI) As stated above, partitioning all node into testing groups will be done by hexagonal tiling and coloring scheme and each hexagonal testing groups conduct sequential GT in order to identify all trigger nodes based on assigned colors. In this section, we will look into those algorithms in detail. 4.3.1 Partition of Nodes Based on Hexagon Tiling and Coloring In this section, we discuss the localized partitioning of the sensor nodes into groups. We consider 2D plane on which the WSN is deployed is partitioned into regular hexagons forming a hexagon tiling, all the nodes located in the same hexagon form a testing group. Let Dh be the diameter of a hexagon as follows: * If 1 < a < 2, then Dh = r SIf 2 < a<3, then Dh= R 2r * If a > 3, then Dh = r In all the above cases, nodes located within the same hexagon will have distance less than or equal to r. Hence, they can communicate with each other in one hop. We now discuss a method which a sensor node can use to locally identify the hexagon it belongs to, thus the group it belongs to. By using Algorithm 4, a node can identify the color of its hexagon and also the time slot assigned to its group in the schedule. We consider that a node v c V knows its neighbors N(v) and using some ad hoc positioning method [48, 49], it can identify its location as (xv, yv) with respect to some reference node. We consider the sink node s c V in the WSN as the reference node such that (xs, ys) = (0, 0). Now, we show that if a node v knows its coordinates (xv, yv) in the Cartesian system then without having the global view of the hexagon tiling, it can locally compute its coordinates (xh, yh) in new coordinate system on the hexagon tiling and further, it can identify the hexagon it belongs to. For instance a node v at coordinates (xv, y,) in the Cartesian coordinates can compute its coordinates (xh, yh) in the new coordinate system as: Xh x 3Yv 3Dh (41) tan 2 yh = y sin / (42) 3 2 The coordinates of the hexagon h(i,j) in which node v is located is given as: y, 3Dh 1 yv / 2 {x (43) i Xv tan  + 2 J= yvsin 3/ D 1+ (44) Now, using the k2coloring algorithm and considering d = R + r, k = [(R + r)/3r = [2(a+1)] (if a < 2 or a > 3) or k = [(R+ r)/ (R2r) 2(a) ] (if 2 < a < 3), node v can compute the color of the hexagon it belongs to as: ColorH(i,)  (j mod k)k + (i mod k) + 1 (45) In order to show the correctness of our method, we prove the following lemmas: Lemma 7. When a > 2, all trigger nodes located in a hexagon h trigger a single common jammer. Proof. Assume that there exists a hexagon h which has two trigger nodes vtl and Vt2 triggering two different jammers ji and j2, respectively. Now, when 2 < a < 3, Dh = R 2r, then max{d(vtl, vt2)} = R 2r and max{d(vtl,J1)} = max{d(vt2,2)} = r, hence, in this case max{d(ji,j2)} = R, which is a contradiction as d(jj2) > R. Similarly, when a > 3, Dh = r, max{d(vtl, vt2)} = max{d(vtl,l)} = max{d(vt2,2)} = r, hence, max{d(j1,j2)} = 3r which is again a contradiction. O Lemma 8. A node v located in a hexagon hi cannot be affected by the Noise injected by a jammerj invoked by a trigger nodes vt located in another hexagon h2, where d(hi, h2) > R + r. Proof. Assume that the Noise injected by the jammerj affects node v, then d(j, v) < R and as max{d(v,j)} = r, so max{d(vt, v)} = R + r, but as d(hl, h2) > R +r, so d(vt, v) > R + r, which is a contradiction. O Lemma 9. Given d = (R + r), the number of colors c use by Algorithm 4 to color the entire hexagon tiling is: ( +a 1) + 1), when Dh r. 1) 2, when Dh = R 2r. Proof. In general, if a hexagon tiling H has hexagons of diameter Dh, then considering a distance d c R+ the k2coloring algorithm needs 2d + l colors, hence, it is straightforward to show that when : * D = rand d = R+ r = (a + )r, k2coloring algorithm needs c = (a + 1) + 1 colors Dh = R 2r= (a 2)rand d= R + r = (a + )r, k2coloring algorithm needs c = 2(_2 + 1)1] colors 73= (a 12 4.3.2 Trigger Nodes Detection Procedure This subsection illustrates how to resort to sequential GT and convexity of trigger nodes so as to identify all trigger nodes with minimum testing latency in each hexagonal testing group. For each node i of hexagonal testing group of colorj, we devised the SGTLTNI algorithm (Sequential Group Testing based Localized Trigger Node Identification) to detect all trigger nodes while all testing groups in the same color can be tested simultaneously. It is reasonable to set each testing round with a predefined constant time slot since each hexagonal testing groups with the same color conduct sequential GT procedure at the same time, and no new testing round would start until activated jammers turn themselves into listening mode. As mentioned, we consider that our network systems are loosely synchronized in the order of seconds. As stated in Lemma 7, based on the constraint on distance between any two jammers ji and j2, d(ji,j2) > R in order to avoid mutual invocation between them, we proved that only one jammer can be activated by nodes within a hexagon. Reactive Jamming Model 4.3.2.1 Sequential group testing based localized trigger node identification (SGTLTNI) algorithm In SGTLTNI algorithm, by using SGT, all trigger nodes can be identified in O( CTI log A) where CT is the convex hull of the set of trigger nodes within a hexagon (see Figure 43) and A is the maximum degree of all nodes in the network, hence, there are no more than A + 1 nodes within a hexagon. We use a method named Quick Identification in order to reduce the number of testing rounds. According to Lemma 7, all triggers within a hexagon activate a same jammer. Thus, a node is a trigger iff it belongs to the intersection of the hexagon with the disk of radius r whose center is the jammer. The convexity of the intersection area guarantee that all nodes within a triangle constructed by three identified trigger nodes are also trigger nodes. In general, all nodes inside the convex hull formed by identified trigger nodes v1, ..., vk are trigger nodes as well without further testing. The complete SGTLTNI algorithm is introduced in Algorithm 5. In every step, T denotes the set of identified triggers, U denotes the set of unidentified nodes. We use ISTN algorithm (presented shortly after) to find among U a single trigger vt that has the maximum distance to the temporary convex hull of T. We show later that vt must belong to the the (final) convex hull CT of all triggers inside the hexagon. We safely eliminate all nodes whose distances from the convex hull of T are larger than that of vt. We also use Quick Identification to include all triggers inside the new convex hull of T u {vt}. The algorithm terminates when all nodes within each hexagon are classified into either triggers or nontriggers. 4.3.2.2 Identification of a single trigger node (ISTN) algorithm The localized ISTN algorithm for identifying one trigger node with maximum 'index' among an ordered set U is illustrated in Algorithm 6. The algorithm works in a same manner with binary search algorithm as it sequentially divides the set into two halves. However, it always tests for the presence of the triggers in the right half first so that if there exist triggers among U, the one with the maximum index (the rightmost trigger) will be returned. ISTN terminates as soon as one trigger node is identified. Clearly, each identification of a trigger node among a set U of nodes by ISTN takes at most log2,( U) rounds. Random Reactive Jamming Model: In case of jammer randomly reacts with probability p, we propose a simple and effective algorithm to identify triggers in Algorithm 7. A set of nodes are identified as nontriggers only if after f testing rounds, no Noise is sensed. We first reveal if there are triggers within the hexagon as in lines 5 to 15. If it is the case, further individual tests are performed to identify whether nodes are triggers. Algorithm 5 SGTLTNI Algorithm 1: INPUT: A set U of sensor nodes inside a hexagon 2: OUTPUT: Set of triggers T in U. 3: All nodes in a group Nh synchronously performs the following to identify all trigger nodes. 4: /* Check for the presence of triggers inside the hexagon */ 5: All nodes in U transmit TEST1 packet in test time 6: if No Noise exists after test + then 7: Return T = 0 and exit 8: end if 9: T0 10: /* Find the first trigger */ 11: vo  ISTN(ConvexHull(U)) 12: U U\ vo, T vo} 13: while U / 0 do 14: for vi U do 15: dT(vi) = Minimum distance from vi to the exterior of the ConvexHull(T) 16: end for 17: Sort U = {v ,..., vlu\} so that dT(vl) < ... < dT(VUI) 18: Find vt  ISTN(U) or return T if no such trigger exists 19: U  U \ {Vt,..., viul} /*Quick Elimination */ 20: T TU {vt 21: T T U { nodes inside ConvexHull(T)} /* Quick Identification*/ 22: U U\ T 23: end while 24: return T 4.4 Theoretical Analysis 4.4.1 Upperbound on Testing Rounds Lemma 10. Trigger identification ends as soon as all triggers in the convex hull CT are identified. Proof. All other triggers are also detected following Quick Identification. One more round of group testing for all the remain nodes will show there are no more triggers among them. D Theorem 4.1. Algorithm 5 requires no more than O( CTI log A) number of rounds where I CTI is the number of vertices on the convex hull of the set of triggers. Algorithm 6 Identification of A Single Trigger Node Algorithm ISTN based on Sequential Group Testing 1: INPUT: U {v v, v2..., v} 2: OUTPUT: vk s.t k = max{i I vi is a trigger} or output no triggers. 3: low = 1, high = m 4: while low < hight do 5: mid= [L(low+ high)] 6: T {Vmid, ... Vhigh} 7: All node in T transmit TESTi packet in test time. 8: if Noise exists after test + time then 9: if (low = high) then 10: Return vlow and exit; 11: else 12: low = mid 13: end if 14: else 15: high = mid 1 16: end if 17: end while 18: return No triggers o Victim Node Trigger Node r *  o Figure 43. Trigger nodes in a hexagon Proof. The key observation is that in every round of the while loop, the chosen vt by ISTN will have the maximum distance to the temporary convex hull of the identified triggers (since the nodes in U are sorted based on the distance to that convex hull). That point must belong to the convex hull CT. Hence, the algorithm loops for at most I CT times and requires at most O( CTI log A) round. O Algorithm 7 FaultTolerance SGTLTNI Algorithm 1: INPUT: A set U of sensor nodes inside a hexagon, an integer f > 0 2: OUTPUT: Set of triggers T in U. 3: All nodes in a group Nh synchronously performs the following to identify all trigger nodes. 4: /* Check for the presence of triggers inside the hexagon */ 5: for i = 1...f do 6: All nodes in U transmit TEST1 packet in test time 7: if No Noise exists after test + then 8: if (i k) then 9: Return T = 0 and exit 10: end if 11: else 12: Break 13: end if 14: end for 15: T 0 16: /* Individual testing */ 17: for all x E U do 18: for i = l...k do 19: Let x transmit TESTi packet in test time and listen 20: if Noise exists no later than test + r then 21: T TU {x} 22: T T U { nodes inside ConvexHull(T)} /* Quick Identification*/ 23: Break 24: end if 25: end for 26: end for 27: return T Since, we can perform testing for all hexagons of same color at the same time, the total number of rounds to identify all triggers in the network will be c x (max{ct} log A 1) where c is the number of colors in Lemma 9 and max{ct} is the maximum size of convex hull of a set of triggers within a hexagon. Although max{ct} may go up to A, the algorithm's performance is often far better than its worst case. 4.4.2 Message Complexity Theorem 4.2. The message complexity of algorithm 5 is O(TA) where T is the number of testing round. Proof. According to our assumption above, since jammers are activated upon receiving a testing message from trigger nodes, the messages required for identification procedure are closely related to testing rounds. Due to the fact that every node within a hexagon does not required additional communication messages except testing messages during running Algorithm 5 the number of testing messages within a hexagonal testing group cannot be more than A in each testing round. The observation underlies on the fact that the maximum diameter of a hexagon is at most r from the Section 4.3. Considering the case that all trigger nodes consist of convex hull within a hexagon, each iteration of testing round will identify at least a trigger node. Hence, the message complexity of algorithm 5 is at most O(TA). D 4.4.3 Random Reactive Jamming Model If triggers are only jammed with probability 1/2 < p < 1, then instead of stopping checking for presence of jammers when no Noise sensed, nodes keep testing up to a maximum number of f times. Since we need at most f rounds to check if a hexagon contains any triggers and at most f(A + 1) rounds for individual testing. The total number of rounds for finding triggers in the network is at most cf(A + 2) where c is the number of colors and f is a predefined parameter. We calculate the expected number of identified triggers within a hexagon. Let d > 0 be the number of triggers within a hexagon (we ignore the hexagons without triggers inside). The probability that some Noise sensed by f testing rounds in lines 5 to 14 in Algorithm 7 is 1 (1 p)f. The probability that a trigger nodes is correctly identified is also 1 (1 p)f (lines 17 to 25). Hence, the expected number of identified triggers will be (1 (1 p)f)(l (1 p)f)d > (1 2(1 p)f)d. To obtain an expected falsenegative rate c i.e. the fraction of triggers that are incorrectly classified as nontriggers, we need to set f = l[ogl_p /2]. For example, if p = 3/4 and the desired falsenegative rate c = 0.01 i.e. 1%, we need f = 4. Note that we only have falsenegative but not falsepositive (nontriggers that are classified as triggers). Algorithm 7 also works for Reactive Jamming Model. Simply, setting p = 1, f = 1 we have an algorithm with the number of rounds is at most c(A + 2). Clearly, the number of rounds does not depends on the size of network (the number of sensor nodes) but on the ratio a = R/r and the maximum degree that is often decided by the density of the network. Hence, the proposed algorithm is scalable for networks of arbitrary size. 4.5 The TNLTCDS Routing Algorithm One of the most important benefits from our decentralized identification procedure is efficient minimization of the damage against reactive jammers by converting trigger nodes into receivers only. By facilitating the shifting procedure, a routing protocol could be constructed in order to avoid the activation of any reactive jammers. In this section, we propose a simple routing algorithm called Trigger Nodes Leaves Tree based on Connected Dominating Set (TNLTCDS). TNLTCDS will neutralize reactive jamming attack as an effective countermeasure. 4.6 Performance Evaluation In this section, an intensive series of simulations has been performed so as to validate the theoretical results and to verify effectiveness of the Localized Trigger Node Identification (LTNI) algorithm against a practical reactive jamming model in general WSNs. In detail, LTNI solution is split into two types of approaches by Sequential Group Testing based LTNI (SGTLTNI) and Individual Testing based LTNI (ITLTNI), and comparison of both schemes will be illustrated to show effectiveness of GT in this section. The performance of the LTNI algorithm against reactive jamming attack is assessed and compared to the performance of JAM [77] approach in terms of latency, message overhead and size of jammed regions (quarantine area) as well. The purpose of these simulations is to validate our approach in various network environments via different team of parameters in network density, quantity of jammers and transmission ratio between jammers and sensors. In addition, the default set of parameters is that n = 3000 nodes and J = 3 jammers are randomly planted using the pseudouniform distribution in a square area of size 1000m x 1000m where transmission range r = 20m and a = R = 3. To investigate the performance of our scheme, the r nodes are varied from n = 1000 to n = 5000 in order to be subjected density of network, at most J = 10 jammers are placed for various size of jammers, and the transmission range of a jammer is varied from a = R = 2.5 to a = = 8. r r Experimental implementation of these simulations do not consider packet losses, linkcongestion or MAC misbehavior except jamming signals in order to evaluate the identification performance only. Since we ran the simulations for each setup 100 times and averaged the results, the results suffice to reflect the efficiency of LTNI algorithms. We modeled the practical networks to validate our algorithm against reactive jammers, one implementation of the reactive jammers, by utilizing ZigBee protocol. A wide range of experiments was conducted based on simple ZigBee protocol using Carrier Sense Multiple Access/Collision Avoidance (CSMA/CA) channel access mechanism. Request To Send (RTS) of size 30 bytes and Clear To Send (CTS) of size 30 bytes are implemented in these experiments. The processing time for any type of messages is uniformly 10ms since sensors have limited resource to generate communication messages, and the propagation speed is 3 x 108 m/s in both algorithms, JAM and LTNI. To communicate with others, every node needs to send at least three messages, a pair of RTS/CTS and a data message and wait the predefined intervals between those messages, 20ms in these experiments in order to simulate practical WSNs. We also assume that the size of a communication message is bigger than 2347 bytes, so that RTS/CTS should be sent before legitimate communications begin. By implementing a real protocol ZigBee, we could report more reliable evaluation of our algorithm based on that. 250 SGTLTNI  ITLTNI  o 200 150 100 50 1000 1500 2000 2500 3000 3500 4000 4500 5000 # of nodes A SGTLTNI  ITLTNI  1 2 3 4 5 6 7 8 9 10 # of jammers B SGTLTNI  ITLTNI  .......... J A ..:::::::: :::.................................. 4AM~" 3 4 5 6 7 8 Figure 44. Rounds by various parameters 4.6.1 Testing Rounds T First of all, the number of testing rounds was measured for a variable size of nodes where n c [1000, 5000] in Figure 4.6 (a), which directly reflects the latency of SGTLTNI algorithms due to the predefined length of a testing round. As shown in Figure 4.6 (a), the testing rounds required to complete the identification of all trigger nodes grow steadily, compared to the incremental size of sensors in denser networks. During the nodes in increments from 1000 until 5000, the testing rounds gradually ascend only around 120 additional rounds in SGTLTNI, but ITLTNI tests at most 150 additional rounds to detect all trigger nodes. That is, the design of SGTLTNI algorithm with Quick 2.2 SGTLTNI  2 ITLTNI  ... JAM .......... .. ... 1.8 ... """" 1.6 1.4 1.2 1000 1500 2000 2500 3000 3500 4000 4500 5000 # of nodes A SGTLTNI ITLTNI I  ... . JAM .... . 1 2 3 4 5 6 7 8 9 10 # of jammers B 3 4 5 6 7 8 Figure 45. Messages by various parameters Identification and Quick Elimination in order to identify all trigger nodes only on convex hull of each hexagon produces a great benefit over the time complexity. Then, the impact to LTNI algorithms from different quantity of jammers are depicted through Figure 4.6 (b) so as to show the effectiveness of our localized scheme in massive jamming disruption. Since r = 20 and a = = 3 with J c [1, 10] are sufficient r conditions to investigate massive jamming attack over n = 3000 in WSNs, no additional jammers beyond J = 10 is simulated. As jammers swell up to 10 times of initial size, just 130 supplementary testing rounds take place in SGTLTNI. Consequently, even with massive impact scenario from large number of jammers against WSNs, the localized identification algorithm produces great robustness and feasibility on practical systems. SGTLTNI ITLTNI ...... , ........ .. **J A M '"""";""": ::" ' 14000 SGTLTNI 25000 SGTLTNI ITLTNI  ITLTNI  12000 JAM ............ 20000 JAM ....... 10000 S 8000 15000 6000 10000 4000 .. 20 5000  i: 2000. 0 0 1000 1500 2000 2500 3000 3500 4000 4500 5000 1 2 3 4 5 6 7 8 9 10 # of nodes # of jammers A B 30000 SGTLTNI  25000 ITLTNI  JAM ............ 2 20000 S15000 10000 5000... ............... 5000 .. ..... 3 4 5 6 7 8 R/r C Figure 46. Runtime by various parameters Finally, the testing rounds shows diversity due to the distance among interferencefree testing groups according to the size of a = R, while parameter a value bigger than 8 would be impractical scenarios. As indicated by Figure 41, disjoint interferencefree testing groups have to be far away at least R + r, therefore the distance between parallel testing groups is tightly related to the number of colors. Due to the fact that bigger a results in more colors with smaller number of interferencefree testing groups, Figure 4.6 (c) discloses increasing trends of tests in both LTNI algorithms. 4.6.2 Message complexity The volume of messages for total identification of trigger nodes is related to energy consumption in resource limited sensor networks, which implies that less messages 180 1 1 1 160 SGTLTNI  . 160 JAM ... .. ......... 140 120 100 1 2 2 80 .,.." 60 ...... ' 40 .."" 20 0 1000 1500 2000 2500 3000 3500 4000 4500 5000 # of nodes A U) _________ U 500 SGTLTNI CU JAM ............ S 400 U 300 0 200 ....... (0 100 # 0 350 SGTLTNI 300 JAM ............ 250 200 150 100 50 0 1 2 3 4 5 6 7 8 9 10 # of jammers B 3 4 5 6 7 8 Figure 47. Nodes in quarantine areas by various parameters result in longer network lifetime generally. The graphs in Figure 4.6 plot the numbers of messages per node from three solutions, JAM, ITLTNI and SGTLTNI so as to report comparative message complexity. Figure 4.6 (a) provides the performance comparison of three approaches in terms of messages per node when changing the network density. The messages from JAM solution are higher than those from both LTNI schemes. In addition, Figure 4.6 (a) shows the superiority of message complexity in LTNI approaches over that in JAM increases as more nodes are placed in networks. The graph explains that 40% more messages are required to construct jammed areas in JAM where n = 1000, however, around 47% less messages suffice to complete the classification of all trigger ..1 ... .. ~~~... . . S......... 100 3500 P = 0.6 '.a................. P = 0.6 (Ind T) ................ 10P = 0.7  3000 P = 0.7 (Ind T)  S 10 P = 0.8  P = 0.8 (Ind T)  S. 0.9 2500 P = 0.9 (Ind T) ......... S 1 P = 1.0 (Ind T)  0.1 2 .. 000 .. S0.1 5 1500 .,.."."" ...... .. 0.01 . 1000 L 0.001 500 . 0.0001 0 1 2 3 4 5 6 1000 1500 2000 2500 3000 3500 4000 4500 5000 # of itertion # of nods Figure 48. The number of rounds T in random reactive jamming model with different values of jamming probability P. nodes from SGTLTNI in total where n = 5000. A part from the number of messages between two approaches, JAM solution demands inevitably more energy than SGTLTNI does since JAM protocol has a couple of different types of communication messages including BUILD and PROBE messages except blind message JAMMED in order to quarantine jammed regions, but SGTLTNI necessitates mainly small size of testing messages to invoke jammers during identification of all trigger nodes. In summary, our localized algorithm is promising approach to apply in practical networks with affordable message complexity and energy consumption. Simulations were also carried out to compare the performance of all three approaches during increasing the number of attackers (jammers) as well as to see how this affects their performances. As revealed in Figure 4.6 (b), both LTNI algorithms significantly outperform JAM approach. Especially, the number of messages from LTNI algorithms is 32 percent less than that of JAM where a single jammer J = 1, and LTNI schemes require 45 percent less than JAM does where J = 10. In addition, When size of jammers is greater than 7, the numbers of messages per nodes obtained from three approaches keep constant trend, since the jammers covered most of nodes in networks. However, when jammers are less than 7, JAM algorithm shows more rapid increasing trend of curve than LTNI algorithms. According to various size of a, comparison of messages per node obtained from ITLTNI and SGTLTNI explain clear improvement over JAM solution. Due to the fact that the message complexity of JAM is considerably dominated by the size of nodes in jammed areas and neighbors of them which need to communicate with others to set up perimeters around the jammed areas, increasing messages from bigger size of adversarial transmission range, larger a, are straight forward. However, importantly, the size of areas with only trigger nodes has nothing to do with a in our approaches, which helps to maintain constant volume of messages. Figure 4.6 (c) shows the messages per node required to complete three algorithms and not only more messages in JAM than in LTNI algorithms but also growing size of messages per nodes in JAM approach. Consequently, even in severe widespread reactive jamming against WSNs, our identification approach validates great robustness and practicality. 4.6.3 Runtime The evaluation of runtime indicates the feasibility of our localized mechanism in various network environments. This observation underlies on the fact that in the research of network security, the prompt recovery is one of key issues to study since it could help to reduce the damage from attacks, so as to maintain stable throughput of networks. In addition, immediate recovery help to keep longer network life time in inherently resourceconstraint and batterylimited WSNs as well. Due to the concept of GT, the runtimes of both LTNI schemes are closely dependent on the number of rounds and the size of nodes within hexagons containing trigger nodes since all the hexagons with trigger nodes require further testing until completion of the identification. However, since a hexagon tiling coloring scheme enables to perform as many parallel testing as possible by utilizing the color assignment generated by k2coloring algorithm in section 4.2.2, the overall runtime for complete identification of all trigger nodes does not drastically grow with the size of jammers or nodes. In Figure 4.6 (a), during the increments of nodes from n = 1000 to n = 5000, our approach delayed only 2 seconds to achieve the total identification of trigger nodes, whereas, JAM takes 7 times as long time as LTNI algorithms do. That is, our identification schemes introduce outperformance on scalability over the JAM. As expected, the runtimes obtained from both LTNI algorithms shows similar increasing trend of curves. Figure 4.6 (b) compares the runtime based on various size of jammers with fixed number of nodes, and SGTLTNI is the best. Particularly, the runtime of SGTLTNI is slightly better than ITLTNI, and the gap between JAM and LTNI approaches get bigger when jammers increase. As described before, bigger a = with fixed number of jammers J = 3 results in r smaller number of interferencefree testing groups, which implies less parallelism on identification consequently. Figure 4.6 (c) plots the time of quarantine procedures, and due to the bigger impact of jammers from bigger a, JAM algorithm requires considerably longer time to quarantine jammed regions when increasing a. In particular, SGTLTNI and ITLTNI shows increasing trend of curves as a growing. Considering the runtime of JAM approach this verification time of identification trigger nodes in LTNI algorithms is quite reasonable. For example, JAM demands 30 seconds to block jammed regions, but SGTLTNI completes the identification of trigger nodes within only 15 seconds where a = 8. In addition, ITLTNI also shows good performance, but higher than SGTLTNI does. Consequently, overall length of runtime depends on the size of nodes in jammed regions, however, hexagon tiling coloring scheme helps to keep small increments of runtime by maximizing parallel testing. 4.6.4 The number of nodes in quarantine areas After constructing a jammingresistant routing path by shifting all identified trigger nodes into receivers, there might be unreachable trigger nodes which are placed too deep to be reached by communication messages. The volume of unreachable trigger nodes could be compared to the number of jammed nodes from JAM algorithm so as to determine the actual size of quarantine areas. The meaning of unreachable trigger nodes in SGTLTNI solution is the same to the jammed nodes in JAM algorithm, since the jammed nodes would not able to receive any messages according to [77], and unreachable trigger nodes also cannot receive any message either. In Figure 4.6, the size of unreachable trigger nodes is always substantially less then the size of jammed nodes from JAM algorithm. Especially, in Figure 4.6 (a) only less than couple of nodes are unreachable trigger nodes and would not be able to receive messages where n c [1000, 5000], but jammed nodes get significantly larger as higher network density. In Figure 4.6 (b), as predicted, the number of unreachable trigger nodes keep small number, less than 10, even in 10 jammers, but jammed nodes sprout with higher population from more jammers in WSNs. With fixed number of jammers and sensor nodes, larger size of a results in bigger impact against WSNs, which implies that more jammed nodes in JAM algorithm. Yet, importantly, our identification approaches will not get affected from a in terms of unreachable trigger nodes. That is, by utilizing the successful identification of all trigger nodes, actual jammed areas in which no node would be able to send out any messages to avoid reactive jamming signal would be very small, so that significantly more nodes would be participated in secure communications than JAM algorithm does. 4.6.5 Random reactive jammers Here the reactive jammers might adopt random responding strategy in order to achieve high stealthy by not jamming some ongoing legitimate communication messages, which makes identification scheme more challenging due to the inaccurate testing outcomes. For this possible attack scenario duplicated testing can be performed to drop the error rate and our scheme still shows robustness against random reactive jamming attack. We shown in Figure 4.6 the falsenegative rate of the Algorithm 7 in a log scale (Recall that we do not have falsepositive). The falsenegative rate linearly decreases in the logscale i.e. exponentially decreases when the number of duplicated testing f increases. Even with the p = 0.6, the falsenegative rate quickly decreases to 1% with f = 6. That is, we only need to repeat the test for 6 times. When we fix the targeted falsenegative rate to be 1%, the number of rounds required is shown in Figure 4.6. The number of rounds increases slightly together with the number of nodes in the network since putting more nodes in a same fixed area rise the density i.e. degree of nodes. However, the rate of increase comes closer to zero when n = 5000. 4.7 Conclusion In this chapter, we devise a novel localized algorithm to efficiently tackle reactive jamming attack problem in WSN by identifying trigger nodes. Our local identification of all trigger nodes achieves high feasibility with low overhead in terms of time and message complexity by leveraging sequential GT and hexagon tiling scheme. We propose the hexagon tiling coloring to exploit the available spacial parallelism to test the WSN for identifying trigger nodes. Based on the color assignment, all testing groups are scheduled to conduct the localized trigger node detection algorithm using the sequential GT Besides the analytical complexity analysis, an intensive series of experiments has shown an outstanding performance of our solution on various WSN settings in terms of scalability and stability. Furthermore, investigation on more stealthy and energy efficient jamming model with simulations indicates robustness and potential of our scheme as well. By embedding identifying the location of jammers based on identified trigger nodes, improved countermeasure for more robust WSNs could be realized since elimination of attackers is inevitably the best way to maintain the soundness of WSNs. CHAPTER 5 CONSTRUCTION OF DOMINATING TREE In this chapter, we study the following new problem, called Dominating Tree (DT), defined as follows. Given an undirected weighted graph G = (V, E, w) representing a wireless sensor network, where V is the set of the nodes in the network, E is the set of all communication links in the network, and w is a nonnegative weight assigned on each edge e = (u, v). We wish to find a Dominating Tree T of G with the minimum weight such that each node in V is either in T or has at least one neighbor in T. The weight of T is defined as the sum of all edge weights in T. 5.1 Overview of Dominating Tree The DT problem has several applications in network design and network routing. For example, multicasting involves the distribution of same data from a central sever to several nodes in the network. Under this setting, we can consider the edge weight as the energy consumption to send a message along that edge. Thus the problem becomes to choose a set of edges (or communication links) of minimum weight for the sever to route the data, which is exactly the DT problem. Since all nodes are at most one hop away from the tree, a message can be first forwarded to the closest node in tree. Then the message can be routed within the DT until it reaches to its destination. In addition, using DT as the routing backbone can help reduce the message overhead complexity. However, no nontrivial approximation algorithm and its hardness were known for the DT problem. From a practical view of energy in WSNs, since the sensor nodes usually have no plugin power, we have to conserve power so that each sensor node can operate for a longer period of time. Many solutions for constructing a routing backbone with minimum energy consumption have been proposed, including the connected dominating set [66]. However, this problem only considers the size of the set obtained, not the weight of edges. In other words, the weight is only associated with each node in these approaches, not with edges where the energy consumption at each link in the communication directly effects the energy consumption of routing. In the theory community, there are some work [6, 72] studying the tree cover problem which is defined as a connected edge dominating set with total minimum edge weights. Arkin et al. first solved this problem in [6]. Later, in [71], this problem can be approximated within a factor of 3 + c. In [72], the author presented a fast combinatorial 2approximation algorithm for the tree cover problem. In contrast to the above work, DT is defined as node dominating sets, not edge dominating sets. Usually, DT always produces a smaller number of links and weight than tree cover, and it is more difficult than tree cover problem. 5.2 Hardness and Approximation In this section, we will show that DT is NPcomplete and it is inapproximated within (1 c) In I V1. Due to the reduction on the proof of NPcompleteness preserving the approximation gap, we only show the reduction in the proof of inapproximability. We then present an approximation framework for the DT problem. 5.2.1 Inapproximability Theorem 5.1. The DTproblem is inapproximable within (1 ) In IVI for any e > 0, unless NP c DTIME(I Vlog log IV) in general graphs. Proof. We will show an approximation ratio preserving reduction from weighted dominating set (WDS) problem to DT WDS asks for the subset of vertices with the minimum weight such that all nodes are dominated. In [25], WDS is shown to be inapproximable within (1 c) In V for any e > 0, unless NP c DTIME(I V0lg log IV). Given an instance of WDS, that is, a graph G = (V, E, w) and a weight function w : V Z+, the problem is to find a minimum weight subset of nodes that dominates all other nodes. We will construct a DT instance G' = (V',E',w') as follows. For each node vi e V, we introduce two new nodes x, and y, shown as gray nodes and black nodes in Figure 51B. Let X = {x,} and Y = {yi}. Let V' = V u X u Y. The set E' and w' are constructed as follows. For all nodes in V, the edges are kept the same. For each edge (vi, vj), let w'(v, vj) = M, where M is a very large number, for instance, we can set M > I VI leE w(e). For all nodes in Y, we introduce a spanning tree on all nodes in Y = {yi,..., yn} with the weight 0 of all spanning edges between yi. For each pair (vi, yi), we create a new edge with weight w'(v,, yi) = w(vi). And finally, to connect the layer V to X, for each node vi e V, we create a new edge (vi, xi) with weight M and edges (vi, xj) with weight M if (vi, vj) e E. Consider a minimum dominating set D e V on G with the minimum weight m, we will show that G' has a dominating tree T with the minimum weight of m. Consider the DT T = D u Dy u Ty, where Dy consists of the set of yi if v, e D, Ty e Y is the nodes spanning all Dy. Since the set D dominates all nodes in X as well, the edges between X and V will not be selected. Notice that M is much larger than w(vi) and 0, the edges between vi e V will not be selected in the dominating tree. In addition, the edges between yi e Y can be arbitrarily selected to form a spanning tree on Dy with no increasing of the minimum weight. Notice that (vi, yi) has to be selected in T to induce the tree, thus the weight of T is m. On the other hand, given DT T of G' with the minimum weight m, we will show that G has a minimum dominating set D of weight m as well. According to the construction, if there exists a node x, e X is selected in T, in order to span this node into a tree, at least one edge with weight M has to be selected. Since we know M is large enough, w.l.o.g, we know M > Zy,1 w(i) will lead to be solution not optimal any more. By construction, there exists an edge (vi, xj) if (vi, vj) c E such that the dominating set D c V can not only dominate all nodes in V but also in X as well. However, in order to dominate the nodes in Y, the edges between vi and yi have to be selected if vi e D. Thus, there is a WDS D with weight m as well, which is optimal. Therefore, DT problem is inapproximable within (1 c) In I VI for any e > 0 unless NP c DTIME(I Vlo09 og0 V) for general graph. The proof is complete. Y \ \ w2(1) , )W(3) W3) (3) W(W(2)2) W ( 4 ) W( 4 i0 0 W(5) W(5) W(6) W (6) X V Y AG B G' Figure 51. Reduction from WDS G to DT G' 5.2.2 Approximating Dominating Tree Before presenting our solution, we first give some basic definitions and lemmas concerning the partial solutions for the DT problem. Then an approximation framework is analyzed with the performance ratio of i(i 1)nll' in time O(n3') for any fixed i > 1, where n is the number of nodes in a given graph. Definition 1. Directed Steiner Tree (DST): In a directed graph G = (V, E) with weight associated with each edge, given a root r e V and a set D c V, the Directed Steiner tree asks us to construct a tree with the minimum weight rooted at r, ensuring that there is at least a path from the root to each node in the set D. We will show that the DT problem can be reduced to the DST problem in polynomial time. And the algorithm for DST can be applied to the DT problem preserving the same performance ratio. For the DT problem, in the graph G = (V, E, w), we introduce a dummy vertex v* for each real vertex v e V, then we add the directional edges from all the neighbors of v (including v itself) to dummy node v*, and we set the weight zero for all these newly added edges. Also for the original edges, we make the edge bidirectional and keep the weight the same as the edge in the original graph. In the new directed graph G'=(V',E',w'), if we pick an arbitrary v e V as a root r, and let all dummy vertices as terminals, we can obtain a DST rooted at r through existing DST algorithm [12] and thus we can obtain an DT of G. It is clear that the reduction is completed in linear time. This reduction is shown as an example in Figure 52. A* B* A B A BF F C C* F C E D E D E* D* AG B G' Figure 52. An example of reduction from G to G' Lemma 11. If r is in the optimal DT then the DT introduced by DST will get the same optimal weight. Also, using the approximation algorithm for DST problem will get the same approximation ratio for the DT problem. Proof: Suppose there is an optimal dominating tree DT*, we can make all the nodes in DT* appear in the DST. Since r is in DT*, for each node in DT, at least one of its neighbors must be in DT, and for any terminal, we can add a directed edge from that neighbor to it. And we know that these edges are zero weighted, so the weight of that DST is w(DT*). Suppose there is an optimal DST ST*, then we have w(DT*) > w(ST*). Also, for the optimal DST, we can eliminate all these zero weighted edges. Since all the terminal nodes have a path from r to them, each node will have at least one of its neighbor appearing in that tree, which means it is a DT So we have w(ST*) > w(DT*). In conclusion, we have w(DT*) = w(ST*). This in turn implies that we can get the same performance ratio for the DT problem as we have for the DST D In [12], the authors provided a polylogarithmic approximation for the DST problem in quasipolynomial time. Then we have an algorithm to obtain this ratio for the DT problem as well: Algorithm 8 Approximation Algorithm for DT 1: INPUT: An undirected weighted general graph G = (V, E, w) 2: OUTPUT: A DT of G 3: Initialize a list to save the DT and its weight 4: Find a vertex u in G with minimum degree 5: Transform G into G' = (V', E', w') by using the transformation technique 6: for each node v that (u, v) c E do 7: Run the DST algorithm [12] in G' and make v as the root r to get a DT 8: Save the DT and its weight in the list 9: end for 10: Return the DT with minimum weight in the list Algorithm 8 is based on the idea of transforming the DT problem to the DST problem, and then use the algorithm in [12] to solve DST problem, thus obtaining the solution for DT Note that after the transformation, we need to find the right root for applying that algorithm, since the ratio for DST is maintained if and only if r is in the optimal DT. This can be done by enumerate the neighbors of the node with the minimum degree, since at least one of the neighbors should be in optimal DT Therefore, from Lemma 11, we can obtain the same approximation ratio for DT. Finally, since the terminal nodes in G' are those dummy vertices which have no outgoing edges, we can simply remove these dummy vertices and those edges incident to them to obtain a DT By setting i = Ig n, the algorithm will obtain an O(lg2 n) approximation in quasipolynomial time which is nO(in) 5.3 Heuristic Algorithm and Analysis In the previous section, we introduced the approximation framework. However, the time complexity is quite high, due to that fact that constructing a DST usually results in a long running time. From this point of view, a heuristic with low time complexity is highly expected. 5.3.1 Algorithm Description We begin this section by defining the terminologies used in our proposed solution as follows. * Leaf edge: An edge is called a leaf edge if it directly connects to a leaf node. * Active edge: An active edge is a leaf edge which links a leaf node from a singlenode subtree to an internal node without considering its weight. Inactive edge: This is an edge connecting two subtrees during the construction of DT. In contrast with active edges, we do consider the weight for this inactive edge. All inactive edges will be edges of the dominating tree. The heuristic algorithm (HeurDT) is depicted in Algorithm 9. From the high level, the algorithm consists of the following main steps: 1. Initialize a tree, where each vertex in a given graph is a separate subtree. 2. Create a sorted list of all edges in G by weights. 3. While all the subtrees are not merged into a tree DT (a) Remove an edge with minimum weight from the sorted as an inactive edge. (b) When the edge from the list connects two different subtrees without any circles. i. Merge them into a subtree by adding the inactive edge to DT. ii. If new internal nodes converted from leaf nodes during merging subtrees have some singlenode subtree neighbors, then link these internal nodes with them by active edges so as to maximize the size of leaf nodes in DT. iii. If the new inactive edge connecting to a node which already has an active edge incident to it, remove this active edge. 4. Prune all the leaf edges of the resultant DT. The details of merging two subtrees are shown in Algorithm 10. The only purpose of active edges is to maximize the number of leaf nodes without considering their weights due to the fact that the total weight of DT does not include the weights from leaf edges. Those edges are active since they might need to be removed when the leaf nodes in active edges have been connected by other inactive edges while merging subtrees. Note that since we do not consider the weights for these active edges, we only keep these active edges as the leaf edges in order to prune them later. All other internal edges must be inactive edges where we do consider the weight during the merging in their increasing order, thus minimizing the weight of DT. An example of removing an active edge during the merging is shown in Figure 53 which depicts the execution of heuristic algorithm. In the Figure 53 (b) and (c), since leaf node g is a new internal node with singlenode subtree i after merging, g links i together by active edge. However, active edge (i, g) has to be removed when new inactive connection (c, i) occurs connecting subtree c to i. Because all internal nodes have to be connected by inactive edges, and if we do not remove active edge (i, g),i will be internal node with an existing active edge (i, g) and thus increasing the weight of DT. Inactive edge Active edge C Leaf node ( Internal node b 8 7 94 8 7 uLinkb ingr e 4nod 7 / 2 \ 9 \ /4 24 \1 \ internal / 2' \ \ / 2 / 2 subtree ito new \ (d) 1 \ 14 (e) 211 4 14 (f) 11 A 4 14 7 6 l10 8 6 10 8 1 2 10 h I 2 h 1 2 hk&2_ \1 2 8 7K b8 c7d b 8 17 SRemove active 2 \ (g) 1 4 14 (h 11 A 4 14 () 4 14 F 8 73 6 10 8 7 6 10 8 110 /\ 1 2\ 5 3 /T 2 4 1ci 2o 9 Figure 53. The execution of HeurDT algorithm Algorithm 9 Heuristic Algorithm for DT (HeurDT) Input: Given an undirected weighted general graph G = (V, E, w) OutputA DT of G DT 0 for each vertex v in a given G do Initialize a subtree Tsub(v)  {v} /* Tsub(v) is the subtree which contains v */ end for Sort all edges in G by weights while the number of subtree Tsub > 1 do e = (u, v)  minimum weight edges from list of sorted edges if u and v are connected by an active edge then convert the active edge into inactive edge else if Tsub(v) == Tsub(u) then continue /* e creates a circle in DT */ else Merge TwoSubtrees(e, DT) end if end while Prune all the leaf edges in resultant DT Return the DT 5.3.2 Runtime Complexity Let n be the number of nodes, m be the number of edges, and A be the maximum degree of G, the runtime of each step is listed as follows: 1. O(n) time to initialize all the nodes. 2. The complexity of Merge Sort algorithm to sort edges by weight in Step 2 is O(mlog m) [17]. 3. In Step 3, for each new internal node u from merging subtrees, all its neighbors need to be checked to see if there are any subtrees with only one node, which is O(A) for each edge to process. However, only merging subtrees would be able to convert leaf nodes into internal nodes, and the total number of merging process takes place at most O(n). As a result, the overall complexity for Step 3 is O(nA). 4. Pruning all leaf nodes only takes linear time. From the above analysis, the total running time complexity is dominated by Step 2. That is, the runtime of this heuristic algorithm is at most O(n2 log n), where the graph is dense enough (A s O(n) and m e n2). Algorithm 10 MergeTwoSubtrees(e = (u, v), DT) Input: (e = (u, v), DT) OutpulDT if u has only one active link in DT then Remove the active links from u in DT Initialize a subtree Tsub(u)  {u} end if if v has only one active link in DT then Remove the active links from v in DT Initialize a subtree Tsub(v)  {v} end if Add e = (u, v) to DT as an inactive link /* Merge two different subtrees */ if u == internal and u has at least one inactive link then for each neighbor node w of N(u) in G do if I Tub(w)l == 1 then Add e = (u, w) to DT as an active link Tsub(u) Tsub(u) U Tsub(w) end if end for end if if v == internal and v has at least one inactive link then for each neighbor node w of N(v) in G do if I Tsub(w) == 1 then Add e = (v, w) to DT as an active link Tub(v) Tsub(v) U Tsub(w) end if end for end if Return DT 5.4 Performance Evaluation In this section, the simulation experiments were conducted to verify the performance of the heuristic algorithm against the optimal solution. In addition, it is easy to see that a minimum spanning tree without leaf (MSTL) of G is also a DT of G, we would like to compare our proposed heuristic to this simple MSTL solution. To evaluate how good the DT is in the WSNs setting, instead of randomly assigning a weight w for each edge, we will consider the energy consumption of each edge during the communication. That is, we consider a disk graph G = (V, E) where each disk represents a transmission range of each node. The weight of each edge (u, v) is defined as w(u, v) = Cv d, where dv is the Euclidean distance between two nodes, u and v, 7 is predefined value to 2 because it is a typical value for an unobstructed environment, and Cv is a random constant. n nodes with transmission range r = 25m are randomly deployed in a predefined area size of 100m x 100m. n varies from 10 to 17 with increment of 1, and 20 network instances were investigated for each value of n, and the results were averaged. IP Optimal ..**.... . HeurAlgo .. MSTL Algo  ....  150 100 50 ..... ..... 0 9 10 11 12 13 14 15 16 17 18 # of nodes A DT Weight 12 IP Optimal ... .....' S Heur Algo  MSTL Algo . .... 8 ) 6 I 2 0 0  9 10 11 12 13 14 15 16 17 18 # of nodes B DT Size 10 1 IP Optimal .... .. .....' 0000 HeurAlgo  MSTL Algo   0000 1000 100   10 9 10 11 12 13 14 15 16 17 18 # of nodes C Running Time Figure 54. Simulation results for HeurDT, MSTL and optimal results First of all, let us present the Integer Programming (IP) formulation of the DT problem. We will use CPLEX to solve this IP and the optimal result will be compared with that of our heuristic. min Y WUvXuv (u,v)EE subject to yu+ yv > 2xv (u, v) E E Sx < S 1 ScV u, vS,(u,v)EE Xv = Yu 1 (51) (u,v)EE uEV SYu > 1 v V uEN+(v) yv E {0, 1} ve V xuv E {0,1} (u, v) E where 1 if vertex v is selected in the optimal solution yv = (52) 0 otherwise 1 if edge (u, v) is selected in the optimal solution Xuv = (53) 0 otherwise In the above IP Formulation, the first constraint shows the relation between vertices and edges in optimal solution, that is, for each edge selected in optimal solution, the end vertices has to be selected; The second and third constraints are similar to the formulation in MINIMUM SPANNING TREE problem to guarantee that the solution is a tree; The fourth constraint shows the constraint of the dominating set. Basically, for each vertex v, at least one of its neighbors or itself has to be selected. Comparing the Weight. Figure 54A illustrates the performance of those three approaches in terms of the weight of DT. As shown in Figure 54A, the DT weights from HeurDT are very close to the optimal solution, which shows HeurDT performs extremely well. In particular, the DT weight from HeurDT has at most 8% of additional weights than the optimal results when n = 15 according to the Figure 54A. In addition, the gap in DT weights between the optimal solution and that of HeurDT does not show growing trends even with increasing number of nodes in G. For example, DT from HeurDT has 8% more weights than optimal DT when n = 15. However, only 7% more weights than optimal weight when n = 17. As expected, the weight from MSTL is much larger than that of HeurDT. This is due to that fact that HeurDT converts all the unnecessary edges into leaf edges except the inactive edges connecting subtrees whereas the typically MST does not consider this. Comparing the Size. As the number of nodes in DT may also affects the performance of any routing protocol based on this virtual backbone, we also would like to compare the performance of these three approaches in term of the DT size. As shown in Figure 54B, the difference in size between optimal size and the DT obtained by HeurDT is very small. According to the Figure 54B, the DT built from HeurDT has at most 1.2 nodes more than the DT from OptDT in the case of the network instance with 15 nodes. Significantly, 54B indicates that the difference in DT size is not affected by network size. However, the size of MSTL is much larger than that of HeurDT. It is very clear to prove that HeurDT maximizing the number of leaves during the merging can drastically reduce the DT size. As n increases, the difference in size of DT between MSTL and the HeurDT significantly increases up, such as 33% additional nodes when n = 4 and 42% more nodes in MSTL when n = 17 than the DT generated by HeurDT This reveals substantial difference of DT size between them when we take the total number of nodes in G into consideration. Comparing the Running Time. Figure 54C presents the running time of all three approaches. As expected, the running time for finding the optimal solution is extremely high. One more time, it confirms that it is too expensive to find the optimal solution, leading to the study of approximation solution. Interestingly, the running time of HeurDT and MSTL is very close to each other. For example, HeurDT takes around 37ms more in average than the MSTL approach. This is due to the time spent on identifying active edges and removing them if they are placed between internal nodes Additionally, converting leaf nodes to internal nodes during merging subtrees necessitates a procedure to check all their neighbors in order to maximize the number of leaf nodes. However, the difference between them is very subtle where as the size and weight of DT obtained from HeurDT is much smaller than that of MSTL. In summary, HeurDT obtains a very good dominating tree with very low time complexity. 5.5 Conclusion In this chapter, we investigate a new NPhard problem of how to construct a dominating tree with minimum weight in WSN. We prove the inapproximability result and provide the approximation framework to solve the DT problem. In addition, due to the high runtime complexity of this approximation algorithm, a much more efficient heuristic algorithm is also proposed. CHAPTER 6 CONSTRUCTION OF VIRTUAL BACKBONE WITH MULTIPLE FACTORS Connected Dominating Set (CDS) has been a well known approach for constructing a virtual backbone to alleviate the broadcasting storm in wireless adhoc networks. Current research has focused on minimizing the CDS size, since computing a minimum size CDS is NPhard. However, little work on CDS with multiple factors constraints has been found in literature. In this chapter, we investigate the tradeoffs among multiple factors in CDS construction, such as fault tolerance, size, diameter and running time. To our best knowledge, no existing research has considered these important factors together in a single model, so that we introduce the multifactors model studying a joint optimization problem in which the objective is to optimize the CDS size, network latency or running time while keeping the fault tolerance. Building on this model, we provide the approximation algorithms with constant ratios. In addition, we present improvement techniques, inspired by the computational geometry and probability, that systematically reduce running time or size of CDS. Simulation results show that our algorithms can gain good tradeoffs among these factors, which coincide with theoretical analysis. 6.1 Overview of Virtual Backbone Many works [35, 1820, 24, 28, 40, 41, 45, 54, 59, 65, 67, 68, 74, 75, 79, 80] seek a minimum size CDS (MCDS),which is NPhard [27], as their major design goal. Minimizing the cardinality of CDS can help to decrease the control overhead since broadcasting for route discovery [33, 51] and topology update [1] is restricted to a small subset of nodes [13]. Therefore broadcast storm problem [73] inherent to global flooding can be greatly decreased. However, there are several important factors that need to be fully investigated. The first important factor is the network latency, also represented as diameter of CDS, which is the longest shortest path between any pair of nodes in CDS. Considering the situation that the receiver is not within the transmission range of the sender, communicate 100 through multihop links by using some intermediate nodes to relay the messages is needed. Since a CDS with large diameter often leads to an increase in propagation error and transmission latency, a CDS with small diameter is certainly preferred for reliable message delivery and short latency. The second important factor is the running time. Due to the frequent link failures and limited power supplies, CDS suffers the challenges from malfunction, hence requiring a new CDS generated by system in a short time, in order to maintain the routing operation in network. Thus, time complexity of the CDS construction is a key factor in the remedy of CDS. Unfortunately, little previous work has measured the running time of their proposed algorithms in simulation. Since low running time of CDS construction is highly preferred in wireless adhoc network, time complexity becomes more important, especially in timesensitive environment. The third important factor is the fault tolerance. As CDS is often very vulnerable due to frequent node failure and link failure, which is inherent in wireless networks, constructing a fault tolerant CDS that continues to function during node or link failure is another issue. The previous work [68, 75, 81, 82] have addressed this issue. However, they only considered the size of CDS together with the fault tolerance, without the diameter of CDS and running time. The main contribution of this work is the multifactors model for a fault tolerant MCDS with bounded network latency (diameter) and the low running time for a feasible solution is expected as well. The characterization of this model is that (1) a variable is involved in this algorithm as an input to make the performance tunable. (2) tradeoffs among multiple factors are shown through analysis, which has not been addresses in related works. More specifically, the proposed progressive algorithm, which is the input of our model, allows for systematic improvement. Taking inspiration from computational geometry and probability, we devise improvement techniques for systematically reducing the running time on locating the center with sacrificing a little performance. In addition, we also present the techniques to remove the redundant nodes for further reducing the size of CDS. In the end, we do simulation to verify of the improvement techniques. The results indicate that the techniques are effective in either reducing the size or the running time. Besides that, we also compare against other recently proposed algorithms, CDSBDD [34], CDSBD [38] and BDA [88] under the same parameters. The results demonstrate our algorithm outperforms CDSBDD, CDSBD and BDA in most testing cases. 6.2 Related Work Algorithms on constructing a CDS can be divided into two categories based on their algorithm designs: centralized algorithms and decentralized algorithms. The centralized algorithms usually yield a CDS with a better performance ratio than that of decentralized algorithms. The decentralized algorithms can be further divided into two categories: distributed algorithms and localized algorithms. In the distributed algorithms, the decision process is decentralized and serialized. In the localized algorithms, the decision process is not only distributed, but also requires only a constant number of communication rounds. Based on the network models, these algorithms can be classified into three types in undirected graph: general graph, Unit Disk Graphs (UDG)[15], where all nodes have the same transmission ranges, and Disk Graphs with Bidirectional Links (DGB) (we will introduce DGB in Section 6.3). 6.2.1 General Graph Several work have been studied in general graph. In [28], two polynomialtime algorithms to construct a CDS is proposed by the authors. The first algorithm has performance ratio of 2(H(5) + 1), where H is a harmonic function and 6 is the maximum degree of G. The idea of the first algorithm is to identify the node with a maximum degree as the root. Then build a spanning tree T at the root, grow T until all nodes are added to T. Then, all leaf nodes are cut off and the remaining nodes in T are a CDS. 102 The second algorithm is a progress of the first algorithm. The second algorithm consists of two steps. The first step is to construct a dominating set and the second step is to connect the dominating set with a Steinter tree. With such improvement, the second algorithm has a better performance factor of H(6) + 2. Later, the two algorithms were simulated by Das et al. in [19, 20, 59]. In [54], Ruan et al. introduced another centralized and greedy algorithm of which the performance ratio is (2 + log 6). Wu and Li [79] proposed an algorithm that can quickly generate a CDS based on the connectivity information within the 2hop neighbors. This approach uses a marking process. In particular, each node is marked true if it has two unconnected neighbors. All the marked nodes form a CDS. The authors also introduced some dominant pruning rules to reduce the size of the CDS. In [74], the authors showed that the performance ratio of [79] is within a factor of O(n) where n is the number of nodes in a network. 6.2.2 Unit Disk Graph In UDG, most of proposed algorithms are to find an MIS and then connect the MIS with minimum number of nodes. In [3, 5, 74], the authors presented a distributed algorithm with a constant performance ratio of 8. Later, Cardei et al. presented another distributed algorithm in [11]. This algorithm has the same performance ratio as previous work. However, the message complexity is lower than that of [74]. As we know that distributed algorithm has a better performance than localized algorithms. In the localized algorithms, in [4], Alzoubi et al. proposed a localized algorithms with a performance ratio of 192. Although the performance of [4] can not compete with that of [74] and [11]. Their algorithm only need one hop neighbors information. Therefore, once a node knows that it has the smallest ID among it neighbors, it becomes a dominator. Then, the dominators can be connected by the intermediate nodes in the next step. In [41], Li et al. proposed another localized algorithm with a performance ratio of 172, which is better than [4]. 103 6.2.3 Disk Graphs with Bidirectional Links Since the specific geographical characteristics of DGB, not all CDS construction algorithms that are applicable in UDG can be applied to DGB. As far as we know, the algorithms in [11, 74] are applicable in DGB. In [67], Thai et al. first proposed the performance ratio of CDS on size in DGB and the two proposed algorithms can be implemented by distributed ways. However, the only difference between two algorithms is the strategy to select MIS, the first algorithm employed Wan's algorithm [74] to choose the nodes in MIS, while the second algorithm used the greedy strategy, that is to include the minimum number of nodes in MIS, thus leading to a better performance than the first algorithm. 6.2.4 Other Results in CDS Mohammed et al. mentioned the problem of constructing CDS with small diameter [45]. However, they did not give a guaranteed performance in their model. In [38], Li et al. studied the CDS problem with bounded diameter in UDG and proposed a constant approximation algorithm, called CDSBD. However, their algorithm is centralized and no experimental results are provided. As an extended work of [38], Kim et al. first made their centralized algorithm to be distributed, then added energy consideration when constructed the CDS. Simulation results and comparison against other recent algorithms were reported at the end. The problem in [34] is that they emphasized that the UDG cannot be used as network models, since the transmission ranges of all nodes may be different. However, they still used UDG as their model through their whole work. In contrast, we will employ a new network model, DGB, to study our model in this work. In summary, none of the previous work have addressed the following issues. First, the performance of our model is tunable, it can be adjusted in a range by an userdefined input. Second, we find out the tradeoffs among CDS size, diameter, running time through theoretical analysis and simulation, that is, it is hard to optimize these factors at the same time. Third, running time is firstly introduced as a metric into the CDS construction and simulation. Fourth, the importance of the center of network is highlighted, since building a CDS rooted at the center will greatly improve the performance. 6.3 Wireless Communication Model and Preliminaries As we described in the above section, in this work, we model the wireless network using a Disk Graph with Bidirectional links (DGB) G = (V, E), which is much more general than UDG. The nodes in V are located in the two dimensional Euclidean plane and each node vi e V has a transmission range r, e [rmi,, rax]. A directed edge (v,, vj) c E if and only if d(v,, vj) < r, where d(v,, vj) denotes the Euclidean distance between vi and vj. Such graphs are called Disk Graphs (DG). An edge (vi, vj) is bidirectional if both (vi, vj) and (vj, vi) are in E, i.e., d(vi, vj) < min{ri, rj}. In this work, we study the multifactors model in disk graphs where all the edges in the network are bidirectional. In this case, G is undirected. Now we give some notations, terminologies and definitions throughout this work. 6.3.1 Notations * r: a root node. * dru: the number of hops in the shortest path between r and u. r R: the transmission range ratio, i.e. R = ma rmin K: the number of independent neighbors of a node u in DGB, if R = 1, then K = 5, otherwise, K = 10([ i( ) + 1)[67], where the independent neighbors of a node u are defined as a set of nodes that adjacent to u satisfying that any two nodes in the set are independent. 6.3.2 Terminologies * u is v's 2hops away neighbor if u and v are not adjacent and they are connected via only one intermediate node. Two nodes x and y are siblings if they are adjacent and dry = dr,. Node u is the parent of node v and v is the child of u such that u and v are connected and drv = dru+1. 105 *Given a root r, node u is a terminal in graph if it has no child. Figure 61 shows an example for the introduced terminologies. Figure 61. Given node r as the root, node a, b, f, g are terminals. r is the parent of c and e. Node a, b are siblings, f, g are siblings. Node a, b, f, g are r's 2hops away neighbors 6.3.3 Definitions * Definition 2. Dominating Set: A Dominating Set (DS) of graph G is a subset C c V such that each node either belongs to C or is adjacent to at least one node in C. A CDS is a DS which induces a connected subgraph. The size of a CDS is the number of nodes in CDS. * Definition 3. Diameter: The diameter of a graph G is equal to the maximum value of dv, where u and v are any nodes in G. Likewise, the diameter of a CDS, normally denoted as d(CDS), is equal to the maximum value of d,, where i,j are any nodes in CDS. * Definition 4. Fault Tolerant MCDS (kmCDS) with Bounded Diameter: Given a DGB G = (V, E) representing a network and two positive integers k and m, find a subset Ckm C V with minimum size, such that: (1) the subgraph induced by Ckm, i.e., G[Ckm], is kconnected, (2) each node not in Ckm is dominated (adjacent) by at least m nodes in Ckm, and (3) the diameter of Ckm is bounded. 6.4 MultiFactors Model and Solutions In this section, we provide a solution for kmCDS with bounded diameter 1 < k < m + 1. First, we need to give the following definitions in graph theory: A graph G is kconnected if it is connected and removing any k 1 nodes from G will not partition G. A separating set or cutvertex of a graph G (V, E) is a set S c V, such that 106 G S has more than one component. When IS = 1, S is a cut vertex. A kblock of a graph is a maximal kconnected subgraph of G that has no separating set. If G itself is kconnected and has no separating set, then G is a kblock. The multifactor model considers three factors (size, diameter and fault tolerance) together. Some recent work [68, 81, 82] has addressed the general fault tolerant MCDS problem. However, none of them mentioned how to bound its diameter. In our previous work[68], we have proposed a solution for kmCDS problem, where 1 < k < m + 1, as illustrated in Algorithm 11. In this chapter, we still use this algorithm to solve our multifactor model. However, a new analysis is proposed for the diameter of kmCDS. The main idea of Algorithm 11 is that merging all the k'blocks in 1Connected mDominating Set (ImCDS) into only one k'block by adding extra nodes, where k'= 2 initially. Then, we increase k' by 1 and repeat the above operation until k'= k. We can use any 1CDS with bounded size and diameter as the input of Algorithm 11. However, in order to make the solution adjustable by the user, an (a, 3)CDS, to be introduced in Section 6.6, is preferred to be an input of Algorithm 11. Algorithm 11 The Solution for Multifactors Model [68] 1: INPUT: A connected DGB G = (V, E) and a CDS C11 with bounded diameter and size 2: OUTPUT: A kmCDS Ckm with bounded diameter 3: Step 1: Based on the input C11, construct a ImCDS Cim by using CDSMIS Algorithm in [68] 4: Step 2: Compute all the k'blocks in Cim, initially k'= 2. 5: Step 3: If there is more than one k'block in Cim, find the shortest path in the original graph that satisfies the two requirements: (i) the path can connect two k'blocks sharing a same separating set to be one k'block of Cim. (ii) the path does not contain any nodes in Cim except the two end points. Then add all intermediate nodes in this path to Cim. 6: Step 4: Repeat Step 2 and 3, until there is only one k'block in Cim. 7: Step 5: Increase k' by 1 and then repeat Step 2, 3 and 4, until k'= k. The resultant Cim will be Ckm. 107 Theorem 6.1. If the input CDS has an approximation ratio of a on size (a > 1), then Algorithm 11 produces a kmCDS with (2K + 2m + l)aapproximation on size, where Ci~ and Cmn are the ImCDS and kmCDS with optimal solution on size respectively. Proof: Ckm is the union of Cim and the nodes added into Cim, in order to make Cim kconnected. The number of nodes we added to make Cim kconnected is at most 2(k 2)( C1m 1)+ 2(K 1)( C1m 1) [68]. Therefore, Ckm =ICm +2(k 2)( Cm 1) 2(K+ 1)( Cm 1) < (2K+ 2k )1 Cim However, in our previous work [68], we already concluded the following inequality, I Cim Thus, I Ckm < (2K 2k 1) Cim <(2K + 2k 1)(K+ m+ a 1)Cm < (2K +2m+ 1)(K m + 1)1 Cml Lemma 12. d(Clm) < d(Cii) + 2. Proof: Since each node not in C11 is dominated by at least one node in C11. Therefore, when we add more nodes into C11 in order to make it to be Cim, we only increase d(C11) by at most 2 hops. o Lemma 13. d(Ckm) < d(Cim) + 2. Proof: Suppose two nodes u and v are in Ckm. The position of node u and v has three possibilities: (1) u, v e C,,. (2) u e Ckm Clm, V Cim. (3) u, v e Ckm Cm. For case (1), the number of hops between u and v is bounded by d(Cim). For case (2), u must be dominated by a node in C,,. Therefore, u is only one hop away from its dominator in Cim and the number of hops between u and v is bounded by d(Cim) + 1. 108 For case (3), u and v are dominated by different nodes in Cm. However, u and v are only one hop away from their dominators. Thus, the number of hops between u and v is bounded by d(Cim) +2. o Theorem 6.2. If the diameter of input CDS is bounded by 3D*, the approximation ratio of the constructed kmCDS on diameter is 3D + 4. Proof: From Lemma 12 and 13, we have the following inequality: d(Ckm) < d(Cim) +2 < d(Cn) + 2 + 2 < D* + 4 < D + 4 6.5 A Better Algorithm for CDS on Diameter Before we introduce the progressive algorithm as the input of Algorithm 11, we would like to introduce an algorithm to determine a CDS with a better approximation on diameter than existing work. As we know, the authors in [38] presented a 3approximation algorithm on diameter, which was the best known result at that time. In this section, an algorithm that guarantees CDS with 2approximation on diameter of CDS is described and the size of CDS is bounded as well. The difference between other existing work [34, 68] and our algorithm is that they try to minimize the CDS size while the diameter is bounded. In contrast, we want to minimize the CDS diameter while the size is bounded. To construct a CDS, we often employ an Maximal Independent Set (MIS) which is also a subset of all the nodes in the network. The nodes in MIS are pairwise nonadjacent and no more nodes can be added to preserve this property. Therefore, each node which not in MIS is adjacent to at least one node in MIS. Thus, an MIS is indeed a DS. If the nodes in MIS are connected by adding more nodes to the MIS, a CDS can be constructed. Here, our algorithm consists of the following three phases, 1. Root r is randomly chosen and we only select the nodes, e.g. node y, into MIS, where dry is an even. Then, the nodes in MIS is colored blue or black, and all other nodes are colored gray or red. 109 2. Swap the colors of blue and red nodes and change some nodes to black according to the rules described in second phase of Algorithm 12. Consequently, the blue nodes and black nodes may be adjacent, but they still dominate other nodes, therefore the MIS with black and blue nodes is changed to a DS. 3. Connect the DS with some intermediate nodes, and DS will be a CDS finally. The details of our algorithm are shown in Algorithm 12. For more clarification, we show an example to illustrate the second phase. Algorithm 12 Algorithm for CDS with 2Approximation on Diameter First Phase: MIS construction 1 : INPUT: A connected DGB G = (V, E) and all nodes are white initially. 2: Randomly choose a root r E V, let {Vk y e V1dry k} and Gk is the subgraph of G induced by Vk, y is in level k, suppose k* is the maximum of k. 3: Find an MIS /2 of G2 by Wan's algorithm. 4: Color all nodes in /2 black and all other nodes gray in G2. 5: for each i, where i = 4, 6, 8, 10...n, if k* is an even, n = k*, otherwise, n = k* 1 do 6: while there exists a white node x in G do 7: if x has a black 2hops away neighbor z in Gi2 then 8: Color x black and all adjacent nodes gray. 9: else 10: Color x blue and all adjacent nodes red if white, otherwise, no color changes. 11: end if 12: end while 13: end for/* from line 5 to 13, we find an MIS with black and blue nodes in Gi, but the process of constructing MIS gives prefer ence to those nodes who has a black 2hops away neighbors in Gi2* 14: while a white node y exists in G do 15: Color y in black and color its adjacent write nodes in gray. /* Note that the node colored black in this loop belong to / for some odd i*/ 16: end while/* All the black and blue node form an MIS. In next phase, we swap the colors of blue and red nodes and change some nodes to black, therefore, the MIS is changed to a DS*/ Second Phase: Swap the color 1: for i 2 to n, where n is an even do 2: for each blue node x in G; do 3: (1) swap the colors of x and x's neighbors only in G 1 (now x is in red and x's neighbor is in blue), (2) suppose in Gi, y is the red node dominated by x, color one y's neighbor only in G_1 black if it is not black. (3) Suppose the red node z in Gi+, is dominated by x, color z black if it is not black. 4: end for 5: end for 6: Color all the blue nodes in G black and all red nodes gray. /* Note: all black nodes at last are a DS, not necessarily an MIS. In next phase we connect all black nodes to be a CDS */ Third Phase: Connect DS to CDS 1: Color r black. 2: for i 2 to k* do 3: for every black node x E G do 4: if x can find a black 2hops away neighbor y in Gi2 then 5: Connect x and y with exactly one node z in Gi1, color z black if it is not black. 6: end if 7: end for 8: end for 9: Let C be the largest black connected component. Note that other than C, each black connected component Ci in Gi for some odd i is not connected with C. But, there must exist one node z in G 1 to connect Ci with C, color z black if it is not black. At last, all Ci in Gi for some odd i are merged into C and only one black connected component exists in DGB finally. 10: The algorithm stops until there is no node changed from gray to black. 11: OUTPUT: All black nodes consist of a MCDS with bounded diameter. 110 Figure 62 presents an example that illustrates the procedure of swapping color in second phase. In this example, we assume that level a + 1 is equal to k*, where a is even, and the MIS with blue and black nodes is produced in first phase. Now, we describe how to swap the colors step by step. 1. The initial situation is shown in Figure 62(a), where black and blue nodes are in MIS. 2. In line 3 of second phase, step (1) is shown in Figure 62(b), the two red nodes in level a 1 are changed to blue, and node x is changed to red. 3. For step (2), in Ga, y is the red node dominated by x, color one y's neighbor p in Ga,_ black. This situation is shown as in Figure 62(c). 4. For step (3), the red node z in Ga+I is dominated by x, color z black. This is shown in Figure 62(d). 5. In line 6, all gray nodes are colored black and all red nodes are colored gray, see Figure 62(e). At this moment, there are only black and gray nodes in the graph and for each black node, we can always find exactly one node that make it connected with other black nodes in upper level within 2 hops, so the number of black node for connecting the DS is at most DSI 1. Theorem 6.3. If TCDS is the CDS determined by Algorithm 12, then I TCDSl < 2(K  1)KCDS*I 1 and d(TcDs) < 2D* + 6 in a DGB. Proof: It is known that for an MIS / in DGB, I/I < K CDS* [67]. However, when we swap the colors of nodes, the MIS is changed to a DS, so, DSI < (K 1)/ since in the worst case, in G,, for even i, if we change the color for each blue node x, we have to change at most K nonblack nodes to black to maintain the whole network dominated by all black nodes, such that x is in MIS and each node in MIS can be adjacent at most K independent neighbors [67]. Therefore, the difference in size between MIS and DS is DSI < (K 1)/ Thus, ITCDSI <_ DSI+ the size of connecting nodes < 21DS 1 < 2(K 1)1/ 1 < 2(K 1)K CDS* 1. For diameter of CDS, suppose black g grey red blue y black gi black P red red y rey grey black red  red x black g level a2 level a1 y level a+1 rey grey black blue blue x black z level a2 level a 1 level a level a+1 black grey black blh 1 7 rey grey black level a2 blue blue ~ level a1 F level a x red level a+1 z grey black ue blue z black z level a2 level a1 level a level a+1 grey black Sp' black level a2 level a1 level a x lb1 k Oa" level a+1 z E Figure 62. An example for second phase in algorithm 12 G has diameter D, then D = k* and the diameter of a CDS is at least D 2 = k* 2. For each black node x in G,, where / is even, x can reach the root with exactly k* hops in TcDS. For odd i, each black node x in G, can reach the root with at most k* + 1 hops in TCDS. Therefore, d( TCDS) < 2k* +2 = (2k* 4)+ 6 = 2D +6. 112 v 0 y 6.6 Progressive Algorithm (PA) and Analysis Although several work [34, 38] can guarantee the CDS size and diameter, their performance ratios are fixed. Considering the flexibility of wireless adhoc network, we introduce (a, 3)CDS into our model so that first, its performance ratios are tunable based on the input. Second, the center of network is involved in the CDS construction to enhance the performance. Third, it can construct a CDS with approximately satisfying the size constraint and diameter constraint. As we intent to balance the size and diameter, the definition of (a, 3)CDS in given in wireless adhoc networks as follows: Definition 5. (a, 3)CDS: Forp/ > 1, a CDS C of G meeting the following two require ments is called an (a, 3) CDS. 1(Size) The size of C is at most a times the minimum CDS size. 2(Diameter) For any pair of vertex u and v in C, d(C) is at most 3 times the mini mum diameter of CDS plus a constant number. In (a, 3)CDS, 3 is an userdefined input, and usually a is a function of 3. Therefore, the value of a depends on the userdefined input P. In the following, we will describe how to generate an (a, 3)CDS, analyze the time complexity and present the tradeoff between the size and diameter through analysis. The general idea of PA is as follows. 1. Root r should locate at the center of network, which is the midpoint of the longest shortest path between two nodes in G. 2. Construct a CDS TcDs rooted at r by using BDA [88] in our previous work, where BDA is an approximation algorithm for CDS with 2Kapproximation on size and 4approximation on diameter. 3. Construct a Shortest Path Tree (SPT) TSPT rooted at r, which only includes all the shortest paths from r to every other node in TCDS 4. Traverse TCDS in a depthfirst manner. When visiting a node u, if the number of hops from r to u in TCDS is larger than a userdefined threshold 3 times the 113 number of hops from r to u in TSPT, then a new path from r to u in TSPT is added in TCDS. If we denote DCDS(U, v) as the number of hops from u to v in TCDS and DSPT(U, v) as the number of hops from u to v in TSPT. The details of PA is as follows: Algorithm 13 Progressive Algorithm (PA) PA (0) 1: Locate the center of network and choose r at the center 2: Build a TCDs by BDA rooted at r 3: Use the algorithm in [46] to construct an SPT TSPT 4: C = FIND (TCDS, TSPT, r, 3) 5: return C FIND (TCDS, TSPT, r, 3) 1: INITIALIZE (TCDS, r) 2: DFS (r) 3: return a desired CDS C INITIALIZE (G, r) 1: for each vertex v e TCDs do 2: d[v] oo 3: r[v]  NIL 4: end for 5: d[r]  0 RELAX (u,v) 1: if d[v] > d[u] + DCDS(U, v) then 2: d[v] = d[u] + DCDS(U, v) 3: [v] u 4: end if DFS (u) 1: if d[u] > /DsPT(r, u) then 2: ADDPATH (u) 3: end if 4: for each child v of u in TCDS do 5: RELAX (u, v) 6: DFS (v) 7: RELAX (v, u) 8: end for ADDPATH (v) 1: if d[v] > DsPT(r, v) and parentspT(v) != NIL then 2: ADDPATH (parentspT(v)) 3: RELAX (parentspT(v), v) 4: end if 1. Root Selection and CDS Tree Construction: With Distributed SPT algorithm [46], each node maintains a global variable, which stores the current longest shortest path in the graph G, if a longer shortest path is found, the global variable of each node will be updated. At the end, we could find the midpoint of global longest shortest path. While constructing TCDS rooted at r by BDA [88], each node u needs to maintain a pointer 7[u] for its parent on the tree TcDs and an upper bound d[u] for the number of hops to r. We use the INITIALIZE and RELAX algorithms in [39] to initialize and maintain both of these attributes. 2. Shortest Path Tree Construction: TsPT rooted at r is constructed by using Distributed SPT algorithm. It only contains all the shortest paths from the root r to every other node in TCDS. 3. Depth First Search (DFS): Traverse the TCDS in a DFS manner beginning from the root r along the paths from r to all the other nodes in TCDS. When node u is reached for the first time, if d[u] is greater than 3 DSPT(r, u), then the shortest Pr, in TSPT is added to TCDS and d[u] and 7[u] are updated. After this, node u's parent v needs to be checked if the updated path from r to u will result in reducing the number of hops from r to v. If so, then v's parent will be checked and so on until the root r is reached. With the execution of BDA, distributed SPT (dSPT) (e.g.[46]), and distributed DFS (dPFS) (e.g.[57]), TCDS, TSPT and a DFS traversal order could be achieved. The details of PA are illustrated in Algorithm 13. To evaluate the correctness of the PA, we examine whether the two constraints in the definition has been satisfied. Taking 3 as an userdefined input, we derive a relationship between a and 3, which shows the relationship between the size of the constructed CDS and the optimal solution of CDS on size. We also analyze the time complexity of the PA. Define w(TcDs) as the total weight of TCDS in G, where we assume each edge has been assigned the unit weight of 1. Then DSPT(U, V) and DCDS(U, v) are equal to the weight of TSPT(U, v) and TcDS(U, v) respectively. Another observation is that STCDS = w( TCDS) + 1, since the number of node in a tree equals to the total number of edges plus 1, which also equals to w(TCDS) + 1. Meanwhile, as we mentioned before, the lower bound of minimum diameter of CDS is D 2. Actually, the upper bound for the minimum diameter of CDS is D, i.e., all the nodes in G are in CDS, therefore, D* = D. 115 Due to the specific structures of CDS, we will classify the following proofs into two cases. case (1): the diameter of SPT T rooted at r that spans all nodes in G is equal to D and all other situations are classified into case (2). Lemma 14. For any pair of nodes u and v in C, the number of hops between u and v is atmost/3(D* +2), when d(T) = D. Proof: When a vertex v is visited, if d[v] > 3DspT(r, v), then shortest path between r and v is added into TCDS by calling ADD PATH. Also, we know that the maximum value for DSPT(r, v) is the height h of TSPT, we will prove that 2h < D* + 2 in the following. After v is visited, d[v] is at most 3DsPT(r, v), which is less or equal to ph and subsequently never increases. For u, the same analysis can also be applied. Therefore, the total number hops between v and u in C is at most 23h, therefore at most 3(D* + 2). Now, we prove that 2h < D* + 2. First, it is easy to see that 2h < d(T) and d(T) = D. Then we have the following: 2h 2 < d(T) 2 < D 2 < D* Therefore, we prove that 2h < D* + 2. O Lemma 15. In case (2), for any pair of nodes u and v in C, the number of hops between u and v is atmost2/3(D* + 1). Proof: If d(T) / D, the worst case is that d( T) = 2D*. A simple example to illustrate that is a ring, the degree of each node in the ring is only 2 and all the nodes in G are included in CDS, see Figure 63. Therefore, h < D* + 1, then the maximum number of hops between u and v is at most 2h, that is 23(D* + 1). E In real wireless adhoc network, case (2) rarely happens, since it requires that all nodes are uniformly deployed as a ring. However, in most cases, they are deployed randomly. Therefore, the diameter of CDS returned by PA is bounded by 3(D* + 2) in most cases. 116 Figure 63. All the nodes in the ring are a CDS with diameter of 8 Lemma 16. The total number of nodes on the added shortest paths is at most (5)K CDS* 3. 81 Proof: Let vo = r and v, v2 .....vk be the vertices that caused shortest path to be added during the traversal, in the order they were encountered. When the shortest path from r to v,(i > 1) was added, the number of hops of the added path was DsPT(r, v,). Also, the nodes on the path to vi has been relaxed in order, so that d[vi] < DsPT(r, vi1) + DCDS(Vi1, i). The shortest path to vi was added because /DsPT(r, vi) < d[v,]. Combining the inequalities, 3DsPT(r, vi) < DspT(r, vi1)+ DCDS(Vi1, Vi) Summing over i bounds from 1 to k, the number of hops of the added paths: k k 3 E DsPT(r, v,) < (DsPT(r v, i) DCDS(Vi1, Vi)) i= 1 i= 1 and therefore k k (3 1) Z DSPT(r, v,) < DCDS(Vi1, v,) i= 1 i= 1 117 The DFS traversal traverses each edge exactly twice, and hence the sum on the righthand side is at most twice w(TcDS), since one hop corresponds to a unit weight of 1, i.e., k SDCDS(Vi1, V) < 2w(TcDs) i= 1 We note that the number of new nodes on the added shortest path is exactly equal to DspT(r, vi) 1 and I TCDS = w(TCDS) +1. Therefore, k (3 1) (DsPT 1)(r, vi) k(3 1) < 2(w(TcDs) 1) 2 i=1 k (3 1) Z(DSPT 1)(r, vi) + k(3 1) < 2 TCDSI 2 i= 1 Here, we intend to maximize k in order to have a tighter bound on Yi(DSPT  1)(r, vi), which is the total number of new nodes on the added shortest paths, Let denote ~1(DSPT 1)(r, i) as Psize for clear representation. Intuitively, k is at most I/, where I is the MIS in TCDS, since all black nodes in MIS of TCDS may cause shortest paths to be added during the traversal. However, the root r and at least two black nodes at level 2 will not cause shortest paths to be added. Therefore, k is at most I/ 3. (/3 1)Psze < 2 TcDS 2 (1/ 3)( 1) < 2(21/ 1) 2 (1 3)( 1) < (5 3)/I + 3/ 7 Since I/I < K CDS* [67], we have: (5 O) K Ps <(5 )KCDS + 3 o 118 Theorem 6.4. Given the value of 3, the approximation ratio a on the size of CDS is ( +3)K 131 Proof: From the above analysis, C is the union of TCDs and the added shortest paths. Therefore, combining the Theorem 3 in [88] and Lemma 16, SC = TCDS + Psize <2K CDS* 1 (5)KCDS*+ 3 < (+3) K CDS* +2 O From the above theorem, we know that the performance ratios depend on the input 3. As the ratio on diameter increases, the ratio on size drops down and vice versa. From a theoretical point of view, we show the tradeoff between size and diameter. Theorem 6.5. The time complexity of the PA algorithm is O(n2), and the message complexity of the PA algorithm is O(n2). Proof: The time complexity and message complexity for BDA are O(n) and O(n log n) respectively [88] and dSPT and dDFS run at most O(n2) time complexity and send O(n2) messages [46] [57]. Now, we analyze the procedure of finding the center of network. The dSPT algorithm is executed at each node x simultaneously, after that, x needs to broadcast the longest path in SPT rooted at x and compare it with the longest paths returned by other nodes. Therefore, this procedure needs 0(n2) time complexity and O(n2) message complexity. Since all other operation only take at most O(n) time complexity and O(n) message complexity, the overall message complexity and time complexity of PA are O(n2) and O(n2). D 6.7 Further Improvements for The Progressive Algorithm The PA in Section 6.6 allows a CDS to be constructed with guaranteed and tunable performance, while the center of network greatly helps to improve the performance, e.g. reducing the size and diameter of CDS, which will be verified in simulation. However, the 119 CDS construction comes at a cost: in PA, we include some redundant nodes that are unnecessary in CDS, and the procedure of locating the center of network is very time consuming, which will be clearly shown in simulation. In this section, we present a set of improvement techniques that systematically remove the redundant nodes and reduce running time with scarifying a little performance. Taking inspiration from probability and computational geometry [44], we present three distinct improvements: reducing multiple paths, removing redundant terminals and locating central area. The first and second techniques reduce the size, whereas the last technique reduces the running time. 6.7.1 Reducing Multiple Paths In PA, after we locate the center r of network, a CDS TCDS rooted at r is determined by BDA, then we test each node, e.g. v, in TCDS to see whether d[v] > /DsPT(r, v), if true, then the shortest path between r and v is added into TcDs by calling ADD PATH. However, there are at least two paths from r to v, one is over TCDs and another one is the shortest path. Now, we can eliminate the path over TCDS to remove the redundant nodes. The technique is: suppose v is at level b, we remove the intermediate node u in TCDS that connects v with the black node in MIS at level b 1. A simple example is illustrated in Figure 64. In Figure 64(a), the black and blue nodes are in TcDs and black nodes represent the MIS. Suppose we set 3 = 1, the red nodes that consist of the shortest path from r to v is added into TCDS. The redundant node in TCDS is u, since v can be connected to the root via shortest path. In Figure 64(b), the redundant node u is removed from TCDS, and the new CDS is smaller than the original one. By doing this, we are able to reduce the CDS size, leading to the low message overhead and transmission error without interfering with CDS diameter. We will show the effectiveness of this technique in simulation. 120 grey red black red grey red black black red x blue vu black blue blue black A grey red black red grey red black black red x blue v u black grey gre black B Figure 64. An example for reducing multiple path 6.7.2 Removing Redundant Terminals Besides reducing multiple paths, we are still able to reduce the CDS size by applying the second technique. Normally, a CDS generated by PA contains some redundant terminals that if we remove them, the modified CDS still respects to the definition of CDS. The main idea of this technique is to remove each terminal in CDS to test if the resultant CDS will violate the definition. If not, then remove the terminal, otherwise, no change is made. Algorithm 14 is shown to implement this technique. For easy understanding, Figure 65 show the situations before and after applying the technique. In Figure 65(a), the black nodes are in TCDS, while the redundant terminal in TCDS is u, since, besides u, x is also dominated by v. In Figure 65(b), the Algorithm 14 Removing Redundant Terminals 1: INPUT: A CDS TCDS with some redundant terminals 2: Color the nodes in TCDS in black and all other nodes in white 3: for each terminal x in TcDs do 4: for each node u e TCDS x do 5: for each u's neighbor v, v TCDs do 6: Color v in red 7: end for 8: end for 9: if all nodes not in TCDs are red then 10: Remove x from TCDS 11: end if 12: Reset all red node to white 13: end for 14: OUTPUT: A CDS with smaller size redundant terminal u is removed from TCDS, and the new CDS is smaller than the original one. X 1 U V r A x U V Figure 65. An example for removing redundant terminals 122 6.7.3 Locating Central Area In PA, we run dSPT on each node in order to locate the center of network. However, the running time is extremely high, and in simulation, we will see the running time increase of two orders of magnitude comparing with BDA. In timesensitive network, this is not acceptable due to the long time consumed on generating a new CDS if the original one fails. Therefore, we present the third technique achieving the comparable running time without sacrificing a lot performance. Before we introduce our third technique, we note that this technique uses the link based diskunion model proposed in [44], each link between two nodes is assigned a weight. Considering a link (u, v) in network, the measure of weight on link (u, v) is given by the number of nodes within the transmission range of nodes u and v (other than u and v). Let D(u, v) denote the disk centered at u and radius Iu, v. Then the weight w(u, v) of the link is defined in term of the number of nodes in disks D(u, v) and D(v, u) as defined below. Definition 6. Weight w(u, v): The weight corresponding to a link (u, v) is given by the count of the nodes in the region given by the union of disks D(u, v) and D(v, u) From the view point of probability, given a randomly generated network, it is highly possible that the center of network is located at the central area of network. In order to have better performance on diameter and size, we usually expect the root of CDS located at the central area, not in the boundary of network. Therefore, we introduce the concept of convex hull in computational geometry [50] to approximate the central area of the given network. We employ a tree and its leaf nodes are defined as the convex hull of the given network. Suppose the root of the tree is in the convex hull, more internal link and very few boundary links are included in the tree. This approach has the drawback of decreasing the probability of the root of CDS located at the central area. To overcome 123 this problem, we can consider the structure of Minimum Spanning Tree (MST), where the weight on each link is defined in Definition 6. Observation: The weight on boundary link is very likely less than the weight on internal link. From the above observation, it is not difficult to see that the central area is more dense (the number of node per area) than the boundary of network. Therefore, the weight on boundary link is normally less than that on internal link. In [44], an O(n2 Ig n) algorithm for constructing Minimum Interference Tree (MIT), which is based on the structure of MST, is reported. This algorithm is a modification of the well known Kruskal's algorithm for constructing MST [17] that increases the probability of boundary links included in MIT. We can use this algorithm to construct our tree and the leaf nodes are defined as the node set of convex hull. Finally, we pick up one node in convex hull as the root of CDS. We can also observe from the tree in Figure 66 that, as expected, most of the links near the boundary of network are presented in our tree, causing most of leaf nodes located at the central area. Figure 66. An example of leaf nodes located at central area. The black nodes consist of the tree. Since there is still small probability that the leaf node is at the boundary, not in central area, if we select such node as the root of CDS, then our technique will fail at the end. In order to minimize the probability of this unexpected situation, we may sort the the leaf nodes based on the degree of each node and then pick up the node with maximum degree as the root of CDS. If there is a tie, a root is selected randomly among those nodes with maximum degree. The reason to do that is the leaf node in central area often has more neighbors than the one at the boundary. To verify the effectiveness of this idea, Figure 67 gives a good example. In Figure 67, the weight of each link is assigned by Definition 6, and the dotted links denote the constructed tree, where a and b are the leaf nodes. If we select b as the root of CDS, then the diameter of CDS will be 5. On the other hand, if a is the root, the diameter will be only 4, since a is in central area, but b is not. Based on our above idea, we will only pick up a as the root since the degree of a is larger than that of b. .0. 4 4. a5 4 4 4 O 5 5,b 4 '.. 6 4 4 5 .. "0' Figure 67. An example of selecting the node with maximum degree as the root of CDS 6.8 Performance Evaluation In this section, we conducted the simulation experiments to measure multiple factors of CDS, e.g. size, diameter, fault tolerance and running time, constructed by our proposed algorithms. First of all, we are interested in comparing the CDSs returned by CDSBDD [34], CDSBD [38], BDA [88] and PA. As far as we know, CDSBDD, CDSBD and BDA are the existing algorithms that guarantee the constant performance ratios on size and diameter, and we will show that PA outperforms CDSBDD, CDSBD and BDA in most testing cases. Second, we also would like to verify the importance of locating the center of network and test the running time of BDA and PA to show the tradeoff between performance and running time. Third, we do experiments by 125 adjusting the userdefined parameter 3 in PA, in order to see how the CDS size and diameter could be balanced. Furthermore, we evaluate the three proposed improvement techniques by comparing PA with improvement against PA without improvement. At last, we test the performance of Algorithm 11 by comparing with PA so that the tradeoff between the fault tolerance and size could be systematically discovered. In [34], the author first proposed the Average Backbone Path Length (ABPL) as another factor to evaluate the CDS. ABPL of a CDS has been defined as the sum of hop distance between any pair of two nodes in CDS divided by the number of all pair of nodes. In our simulation, we evaluate ABPL in addition to diameter and size, since the diameter only represents the worst case path length of CDS, ignoring the average path length, while ABPL captures average path length for message delivery. Therefore, it is our interests to measure the ABPL of CDS. To simulate the network, we randomly deployed n nodes to a fixed area of 1,000m x 1,000m. n changed from 10 to 100 with an increment of 5. Each node vi randomly chose the transmission range r, e [rmi, rmax] where rin = 100m and rmax = 200m. For each value of n, 1,000 network instances were investigated and the results were averaged. 6.8.1 Performance for CDSBDD, CDSBD and PA We conducted simulations to compare the performance of CDSBDD, CDSBD and PA. CDSBD is a centralized algorithm proposed in [38]. It selects a root randomly and spans a CDS from the root. The approximation ratios of CDSBD are 11.4 and 3 on size and diameter respectively. On the other hand, the authors in [34] proposed CDSDBD that can be implemented in distributed way, however, it only guarantees 4approximation on diameter, but 6.906approximation on size. For the purpose of fairness, we set3 = 3 (the approximation ratio of PA on diameter) in PA. Figure 68A shows that the diameters of CDS built by the three algorithms are easy to distinguish, since the gap is clear to observe. Also, we notice that PA outperforms 126 16 CDSBD / 60 CDSBD * 14 CDSBDD CDSBDD  12 PA 50 PA  0 10 1 40 Number of nodes in the network Number of nodes in the network 8 0 8) 30 E 6 (Cl PA 20 4 2 10 0 0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network Number of nodes in the network A Compare the diameter of CDS B Compare the size of CDS F CDSBD, CDSBDD and PA performs better than CDSBDD 4 0 3 1 0 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network C Compare the ABPL of CDS Figure 68. Performance for CDSBD, CDSBDD and PA CDSBDD because of the lower approximation on diameter in PA, and CDSBD also performs better than CDSBDD. Figure 68B provides the performance comparison of the three algorithms on the size of CDS. It shows PA always constructs a CDS with smaller size than CDSBD and CDSBDD, which is much better than theoretical analysis we gave in Section 6.6. As expected, it is reasonable that CDSBDD performs better than CDSBD, since CDSBD adds more nodes in CDS to shorten the diameter, which will cause the increase on size, but the gap between the two curves is not large. Therefore, the performance of PA is satisfactory on CDS size. In Figure 68C, as the number of nodes in network increases, the CDS returned by PA always has a lower ABPL than other two CDSs determined by CDSBDD and 127 CDSBD. Since the better performance of PA on diameter and size, PA is superior to other algorithms in terms of ABPL. 6.8.2 Performance for BDA and PA 60 5 BDA BDA  BDAMid BDAMid 0 3 5 40 6 303 E 2 i:5 20 10 00 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network Number of nodes in the network A Compare the diameter of CDS B Compare the size of CDS 4 BDA * 3 5 BDAMid 3 PA  3 2 25 2 2 1 15 1 05 0 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network C Compare the ABPL of CDS Figure 69. Performance for BDA and PA The purpose of this simulation is to verify the importance of root selection and the tradeoff between performance and running time at the same time. In order to highlight the root selection, we use a variation of BDA, called BDAMid, as a reference. Compared to BDA, BDAMid selects the center of network as the root instead of choosing randomly. Also, we include PA in this simulation and is set to 1. Figure 69A compares the diameter of CDS constructed by the three algorithms. It is shown that, under different number of nodes deployed in networks, the CDS built by PA has the smallest diameter. We observe that the gap between BDA and BDAMid 128 is shown clearly, which indicates that the CDS could achieves smaller diameter with locating at the center of network. On the other hand, the difference between BDAMid and PA is small, which highlights an important fact that if the center of network is detected, the diameter of CDS rooted at the center will be nearly optimal, even using an algorithm that only guarantees a loose bound on diameter, such as BDA. In order to see how far the diameter of CDS returned by BDAMid from the optimal solution, we set 3 to 1 in PA. Since with P = 1, PA will produce a CDS with minimum diameter in most cases. In Figure 69B, we present the size of CDS obtained from all three algorithms, depending on the number of nodes deployed. The sizes of CDSs returned by the three algorithms are close to each other and they all increase with the number of nodes. Also, considering the same number of nodes, BDA returns a larger size of CDS than PA and BDAMid, although the gaps between these algorithm look small in Figure 69B. This illustrates that the size of CDS can be reduced by choosing the center of network as the root. As shown in Figure 69C, PA achieves a CDS with smallest ABPL, whereas BDAMid still performs better than BDA. Overall, PA leads the performance on size, diameter and ABPL due to the center of network. Therefore, it appears to be an important issue in the construction of CDS. Table 61 summarizes the running time under different number of nodes. As the complexity analysis indicates, the runtime of BDAMid and PA is much longer than that of BDA. This is due to the long time spent on detecting the center of network. Moreover, we show in Table 61 that the BDAMid still runs faster than PA, since PA needs to compute TSPT to shorten the diameter. When the number of nodes increases, PA and BDAMid spend more time on detecting the center of network. Therefore, it is a tradeoff between the size/diameter of CDS and running time. 129 Number BDA BDAMid PA PA 3rd Tech. of Node Runtime Runtime Runtime Runtime 10 0.0001 0.0003 0.0006 0.0003 15 0.0002 0.0012 0.0014 0.0003 20 0.0003 0.0044 0.0046 0.0004 25 0.0004 0.0108 0.0132 0.0032 30 0.0006 0.0236 0.0280 0.0056 35 0.0007 0.0442 0.0536 0.0108 40 0.0012 0.0768 0.0980 0.0232 45 0.0012 0.1222 0.1558 0.0352 50 0.0013 0.1836 0.2372 0.0556 55 0.0014 0.2812 0.3742 0.0956 60 0.0030 0.4016 0.5418 0.1438 65 0.0036 0.5414 0.7378 0.2010 70 0.0040 0.7552 1.0332 0.2832 75 0.0046 0.9842 1.3488 0.3712 80 0.0046 1.3050 1.8086 0.5110 85 0.0050 1.6676 2.3030 0.6450 90 0.0045 2.1294 2.9578 0.8400 95 0.0048 2.6672 3.6630 1.0078 100 0.0060 3.7564 5.2224 1.4856 Table 61. Runtime(ms) 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network A Compare the diameter of CDS 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network B Compare the size of CDS Figure 610. Performance based on different 3 6.8.3 Performance Based on Different 3 In the above simulations, 3 is fixed. Now we conduct the simulations with different values of 3. We study the relationship between 3 and the size of CDS and the relationship between 3 and diameter of CDS. Results are shown in Figure 610. 130 In Figure 610A, each line represents the diameter of CDS based on one of different values of 3. When 3 is set to 1, PA adds a shortest path from v to r if DcDs(r, v) is larger than DSPT(r, v). Therefore, PA with 3 = 1 returns a CDS with the smallest diameter. When P is set to 4, PA only adds the path from v to r in TCDS under the condition that DcDs(r, v) is greater than 4 times of DSPT(r, v). Thus, the CDS by PA with 3 = 4 has the largest diameter. For 3 = 2, the corresponding line is in the middle. Therefore, as we expected, the diameter of CDS built by PA could be controlled by adjusting the values of 3. In Figure 610B, each line represents the size of CDS based on one of different values of 3. When 3 is set to 1, if DcDs(r, v) is larger than DsPT(r, v), PA adds a shortest path from v to r. This strategy will incur more nodes to be added. On the opposite, when 3 is set to 4, PA results in a CDS with smaller size. For 3 = 2, the corresponding line is in the middle, the same situation as in Figure 610A. In conclusion, the performance of PA can be balanced depending on the value of 3 and the tradeoff between size and diameter is clear. 6.8.4 Performance for Improvement Techniques 5 50 4 40 3 30 10 2 2 20 PA with 1st and 2nd tech PA with 1st and 2nd tech  PA without 1st and 2nd tech PA without 1st and 2nd tech 0 0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network Number of nodes in the network A Compare the diameter of CDS B Compare the size of CDS Figure 611. Performance for the first and second improvement techniques In this section, we are interested in evaluating the effectiveness of the presented improvement techniques. Since the first and second techniques are devoted into reducing the size of CDS, we would like to test them together, whereas the third technique is performed alone. Figure 611A describes the performance in terms of the diameter of CDS. As we expected, the diameter is not affected by the first and second techniques. Meanwhile, as observed from Figure 611 B, the size of CDS is reduced when the two techniques apply. Clearly, we believe that the first and second improvement techniques are effective in reducing the size of CDS. 6 5 50 S4 40 04 3 30 2 N . .) 20 1 10 PAwith3rdtech PA with 3rd tech 0 PA without 3rd tech PA without 3rd tech 0 0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network Number of nodes in the network A Compare the diameter of CDS B Compare the size of CDS Figure 612. Performance for the third improvement techniques Table 61 also summarizes the comparison of runtime by PA with and without the third improvement technique. Incredibly, the third technique achieves the reduction of running time greatly, although here it sacrifices a little performance on size and diameter, as shown in Figure 612A and 612B, which indicates the tradeoff between running time and size/diameter. However, it is still promising to find the central area in network, in order to achieve the fast construction of CDS. 6.8.5 Performance for MultiFactors Model We are also interested in evaluating the performance of Algorithm 11. We intend to illustrate that Algorithm 11 improves the fault tolerance of CDS by adding marginal overhead (in terms of the number of nodes added into CDS). We take the CDS generated by PA as the input of Algorithm 11, and we set k = 2, m = 1 and 3 = 2. 132 4U 0 3 / 30 2 320 1 10 Algorithm 1 Algorithm 1  PA PA 0 0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 Number of nodes in the network Number of nodes in the network A Compare the Diameter of CDS B Compare the Size of CDS Figure 613. Performance for multifactors model Figure 613A compares the performance of Algorithm 11 and PA in terms of the diameter of CDS. As we expected, there is little difference on the diameter of CDS based on the two algorithms, which perfectly matches our theoretical analysis for the diameter of kmCDS. Therefore, Algorithm 11 enhances the fault tolerance of CDS without affecting its diameter greatly. Meanwhile, as observed from Figure 613B, the size of kmCDS obtained from Algorithm 11 is certainly larger than CDS by PA. Specifically, the performance of the two algorithms is relatively proportional. As observed from our experiments, the size of kmCDS obtained from Algorithm 11 is almost 1.1 times the size of CDS returned by PA. The results indicate that considering the fault tolerance will increase the size of the CDS at the same time. However, the increase in size is still bounded and predictable. Therefore, it is clear to see the tradeoff between the fault tolerance and size. 6.9 Conclusions In this chapter, we introduce the multifactors model studying a joint optimization problem in which the objective is to reduce the CDS size, network latency and running time while keeping the fault tolerance of CDS as well. Building on this model, we provide an approximation algorithm with constant ratios as the solution. After that, an algorithm for CDS construction that guarantees the best approximation on diameter 133 is provided. More importantly, the progressive algorithm, featured with a tunable and outstanding performance, is addressed in details as an input of our model. In addition, we present improvement techniques for progressive algorithm that systematically reduce running time or size of CDS. Simulation results show that our algorithms can gain good tradeoffs among these factors, which coincide with theoretical analysis. Many research problems remain open. Our simulation is preliminary, and stronger results may allow us to better predict and find out the tradeoffs among the factors of CDS. A better understanding of probability and graph theory may yield localized algorithms and better performance. But, even with our current model, theoretical analysis and simulation show large improvements over previous solutions. CHAPTER 7 CONCLUSION This dissertation investigates a novel approach against the reactive jamming attack by identifying triggers and building virtual backbone based on the detection. In other words, the proposed approach consists of two main parts including identification of all trigger nodes to quarantine jammed areas and construction of virtual backbone to deroute the region. First of all, two identification schemes are introduced, centralized and decentralized algorithms. The proposed centralized identification procedure shows several benefits of detecting trigger nodes in practical multipleradio WSNs. By utilizing nonadaptive group testing scheme, disjoint disk cover method, and cliquebased clustering, this countermeasure achieves low overhead in terms of time and message complexity, thus is practical for general WSNs. Also this work proposed the first localized algorithm to identify triggers in networks. Due to its localization and simplicity the algorithm is wellscalable. In addition, to against more sophisticated jammers that randomly react to signal we also devise an efficient faulttolerance algorithm in both algorithms. Secondly, in order to construct effective virtual backbone not only for jamming resistant routing protocol but also for efficient broadcasting/multicasting routing, this dissertation investigates dominating tree and multifactor model with approximation algorithms in wireless adhoc networks. DT problem can help each node in the network to construct its own broadcast tree with small amount of message overhead. Furthermore, since there are several important factors that need to be fully investigated for broadcast latency and faulttolerant routing protocol in the networks, this dissertation designed multifactor models for a fault minimum size CDS with bounded network latency and the low running time for a feasible solution. 135 Theoretical analysis and simulation results endorse the scalability and efficiency of our new approach against reactive jamming attack in terms of latency and message complexity. 136 REFERENCES [1] Optimized Link State Routing Protocol for Adhoc Networks. 2002. [2] Acharya, Mithun and Thuente, David. "Intelligent Jamming Attacks, Counterattacks and Counter Attacks in 802.11 b Wireless Networks." Proceedings of OPNET WORK. 2005. [3] Alzoubi, Khaled M., Wan, PengJun, and Frieder, Ophir. "Distributed Heuristics for Connected Dominating Sets in Wireless Ad Hoc Networks." Journal of Communica tions and Networks 4 (2002): 2229. [4] Alzoubi., Khaled M., Wan, PengJun, and Frieder, Ophir. "MessageOptimal Connected Dominating Sets in Mobile Adhoc Networks." MobiHoc '02: Pro ceedings of the 3rd ACM international symposium on Mobile ad hoc networking & computing. New York, NY, USA: ACM, 2002, 157164. [5] Alzoubi, Khaled M., Wan., PengJun, and Frieder, Ophir. "New Distributed Algorithm for Connected Dominating Set in Wireless Ad Hoc Networks." HICSS '02: Proceed ings of the 35th Annual Hawaii International Conference on System Sciences (HICSS'02)Volume 9. Washington, DC, USA: IEEE Computer Society, 2002, 297. [6] Arkin, Esther M., Halldorsson, Magnus M., and Hassin, Refael. "Approximating The Tree And Tour Covers Of A Graph." 1993. [7] Baruch, Awerbuch, Andrea, Richa, and Christian, Scheideler. "A JammingResistant MAC Protocol for Singlehop Wireless Networks." PODC '08: Proceedings of the twentyseventh ACM symposium on Principles of distributed computing. New York, NY, USA: ACM, 2008, 4554. [8] Bellardo, John and Savage, Stefan. "802.11 DenailofService Attacks: Real Vulnerabilities and Practical Solutions." Proceedings of the 12th conference on USENIX Security Symposium. 2003. [9] Bomze, Immanuel M., Budinich, Marco, Pardalos, Panos M., and Pelillo, Marcello. "The Maximum Clique Problem." Handbook of Combinatorial Optimization. Kluwer Academic Publishers, 1999, 174. [10] Bron, Coen and Kerbosch, Joep. "Finding All Cliques of an Undirected Graph." Commun. ACM 16 (1973).9: 575577. [11] Cardei, Mihaela, Cheng, Xiaoyan, Cheng, Xiuzhen, and zhu Du, Ding. "Connected Domination in Multihop Ad Hoc Wireless Networks." In Proc. the 6th Interna Conference on Computer Science and Informatics. 2002. [12] Charikar, Moses, Chekuri, Chandra, yat Cheung, To, Dai, Zuo, Goel, Ashish, Guha, Sudipto, and Li, Ming. "Approximation Algorithms for Directed Steiner Problems." Journal of Algorithms. 1998, 7391. 137 [13] Cheng, Xiuzhen, Ding, Min, Du, Hongwei, and Xiaohua, Jia. "On the Construction of Connected Dominating Set in Adhoc Wireless Network." in Special Issue on Ad Hoc Wireless Communications and Mobile Computing. 2004. [14] Chiang, Jerry T. and Hu, YihChun. "Dynamic Jamming Mitigation for Wireless Broadcast Networks." INFOCOM 2008. The 27th Conference on Computer Communications. IEEE. 2008, 12111219. [15] Clark, Brent N., Colbourn, Charles J., and Johnson, David S. "Unit Disk Graphs." Discrete Math. 86 (1990).13: 165177. [16] Codenotti, Paolo, Sprintson, Alexander, and Bruck, Jehoshua. "AntiJamming Schedules for Wireless Data Broadcast Systems." Information Theory, 2006 IEEE International Symposium on (2006): 18561860. [17] Cormen, Thomas H., Leiserson, Charles E., Rivest, Ronald L., and Stein, Clifford. Introduction to Algorithms. The MIT Press, 2001, 2nd revised edition ed. [18] Dai, Fei and Wu, Jie. "On Constructing kConnected kDominating Set in Wireless Networks." In Proceedings of the 19 th International Parallel and Distributed Processing Symposium (IPDPS). 2005. [19] Das, Bevan and Bharghavan, Vaduvur. "Routing in AdHoc Networks Using Minimum Connected Dominating Sets." International Conference on Communica tions. 1997, 376380. [20] Das, Bevan, Sivakumar, Raghupathy, and Bharghavan, Vaduvur. "Routing in Ad Hoc Networks Using a Spine." International Conference on Computers and Communication Networks. 1997, 120. [21] Desmedt, Yvo, SafaviNaini, Rei, Wang, Huaxiong, Batten, Lynn, Charnes, Chris, and Pieprzyk, Josef. "Broadcast Antijamming Systems." Networks, 1999. (ICON '99) Proceedings. IEEE International Conference on. 1999, 349355. [22] Du, DingZhu and Hwang, Frank K. Combinatorial Group Testing and its Applica tions(2nd ed.). World Scientific, Singapore, 1999. [23] Du., DingZhu and Hwang, Frank K. Pooling Designs: Group Testing in Molecular Biology. World Scientific, Singapore, 2006. [24] Du, DingZhu, Thai, My T., Li, Yingshu, Liu, Dan, and Zhu, Shiwei. "Strongly Connected Dominating Sets in Wireless Sensor Networks with Unidirectional Links." APWeb. 2006, 1324. [25] Feige, Uriel. "A Threshold of Ln N for Approximating Set Cover." Journal of the ACM 45 (1998): 314318. 138 [26] Felstead, Barry E. "Follower Jammer Considerations for Frequency Hopped Spread Spectrum." Military Communications Conference, 1998. MILCOM 98. Proceedings., IEEE. vol. 2. 1998, 474478. [27] Garey, Michael R. and Johnson, David S. "Computers and Intractability. A guide to the Theory of NPcompleteness." New York, NY, USA: Freeman, 1979. [28] Guha, Sudipto and Khuller, Samir. "Approximation Algorithms for Connected Dominating Sets." Algorithmica 20 (1996): 374387. [29] Gupta, Rajarshi and Walrand, Jean. "Approximating Maximal Cliques in Adhoc Networks." Personal, Indoor and Mobile Radio Communications, 2004. PIMRC 2004. 15th IEEE International Symposium on 1 (2004): 365369. [30] Hang, Wang, Zanji, Wang, and Jingbo, Guo. "Performance of DSSS against Repeater Jamming." Electronics, Circuits and Systems, 2006. ICECS '06. 13th IEEE International Conference on (2006): 858861. [31] Hassan, Amer A., Hershey, John E., and Schroeder, James E. "On a Follower ToneJammer Countermeasure Technique." Communications, IEEE Transactions on 43 (1995).234: 754756. [32] Hassan, Amer A., Stark, Wayne E., and Hershey, John E. "FrequencyHopped Spread Spectrum in the Presence of a Follower Partialband Jammer." Communi cations, IEEE Transactions on 41 (1993).7: 11251131. [33] Johnson, David B. and Maltz, David A. "Dynamic Source Routing in Ad Hoc Wireless Networks." Mobile Computing. Kluwer Academic Publishers, 1996, 153181. [34] Kim, Donghyun, Wu, Yiwei, Li, Yingshu, Zou, Feng, and Du, DingZhu. "Constructing Minimum Connected Dominating Sets with Bounded Diameters in Wireless Networks." IEEE Transactions on Parallel and Distributed Systems 20 (2009): 147157. [35] Law, Yee Wei, Hartel, Pieter, Den, Jerry Hartog, and Havinga, Paul. "Linklayer Jamming Attacks on SMAC." Wireless Sensor Networks, 2005. Proceeedings of the Second European Workshop on. 2005, 217225. [36] Levitt, Barry K. "FH/MFSK Performance in Multitone Jamming." Selected Areas in Communications, IEEE Journal on 3 (1985).5: 627643. [37] Li, Mingyan, Koutsopoulos, lordanis, and Poovendran, Radha. "Optimal Jamming Attacks and Network Defense Policies in Wireless Sensor Networks." INFOCOM 2007. 26th IEEE International Conference on Computer Communications. IEEE. 2007, 13071315. [38] Li, Yingshu, Kim, Donghyun, Zou, Feng, and Du, DingZhu. "Constructing Connected Dominating Sets with Bounded Diameters in Wireless Networks." 139 Wireless Algorithms, Systems and Applications, International Conference on 0 (2007): 8994. [39] Li, Yingshu, Thai, My T., Wang, Feng, and Du, DingZhu. "On the Construction of a Strongly Connected Broadcast Arborescence with Bounded Transmission Delay." IEEE Transactions on Mobile Computing 5 (2006).10: 14601470. [40] Li, Yingshu, Thai, My T., Wang, Feng, Yi, ChihWei, Wan, PengJun, and Du, DingZhu. "On Greedy Construction of Connected Dominating Sets in Wireless Networks: Research Articles." Wirel. Commun. Mob. Comput. 5 (2005).8: 927932. [41] Li, Yingshu, Zhu, Shiwei, Thai, My T, and Du, DingZhu. "Localized Construction of Connected Dominating Set in Wireless Networks." NSF International Workshop on Thoretical Aspects of Wireless ad hoc, Sensor and PeertoPeer Networks. 2004. [42] Ling, Qi, Li, Tongtong, and Ding, Zhi. "A Novel Concept: Message Driven Frequency Hopping (MDFH)." Communications, 2007. ICC '07. IEEE Interna tional Conference on. 2007, 54965501. [43] Ma, Jianqing, Zhong, Yiping, and Zhang, Shiyong. "FrequencyHopping Based Secure Schemes in Sensornets." CIT '05: Proceedings of the The Fifth Interna tional Conference on Computer and Information Technology. Washington, DC, USA: IEEE Computer Society, 2005, 459463. [44] Martin, Burkhart, Pascal, Von Rickenbach, Roger, Wattenhofer, and Aaron, Zollinger. "Does Topology Control Reduce Interference?" MobiHoc '04: Pro ceedings of the 5th ACM international symposium on Mobile ad hoc networking and computing. New York, NY, USA: ACM, 2004, 919. [45] Mohammed, K., Gewali, L., and Muthukumar, V. "Generating Quality Dominating Sets for Sensor Network." ICCIMA '05: Proceedings of the Sixth International Con ference on Computational Intelligence and Multimedia Applications. Washington, DC, USA: IEEE Computer Society, 2005, 204211. [46] Mu, Fi, Brim, Lubos, Cerna, Ivana, Krical, Pavel, and Pelanek, Radek. "Distributed Shortest Path for Directed Graphs with Negative Edge Lengths." Tech. rep., Masaryk University, 2001. [47] Navda, Vishnu, Bohra, Aniruddha, and Ganguly, Samrat. "Using Channel Hopping to Increase 802.11 Resilience to Jamming Attacks." INFOCOM 2007. 26th IEEE International Conference on Computer Communications. IEEE. 2007, 25262530. [48] Niculescu, Dragos and Lab, Dataman. "Ad hoc positioning system (APS) using AOA." INFOCOM 2003. TwentySecond Annual Joint Conference of the IEEE Computer and Communications Societies. IEEE. vol. 3. 2003, 17341743. [49] Niculescu, Dragos and Nath, Badri. "Ad Hoc Positioning System (APS)." GLOBE COM2001. Nov 2001. 140 [50] O'Rourke, Joseph. Computational Geometry in C (Cambridge Tracts in Theoretical Computer Science). Cambridge University Press, 1998. [51] Perkins, Charles and Royer, Elizabeth. "Adhoc OnDemand Distance Vector Routing." In Proceedings of the 2nd IEEE Workshop on Mobile Computing Systems and Applications. 1997, 90100. [52] Perrig, Adrian, Stankovic, John, and Wagner, David. "Security in Wireless Sensor Networks." Commun. ACM 47 (2004).6: 5357. [53] Popper, Christina, Strasser, Mario, and Capkun, Srdjan. "JammingResistant Broadcast Communication without Shared Keys." ETH Zurich DINFK Technical Report 609. 2008. [54] Ruan, Lu, Du, Hongwei, Jia, Xiaohua, Wu, Weili, Li, Yingshu, and Ko, Kerl. "A Greedy Approximation for Minimum Connected Dominating Sets." Theor. Comput. Sci. 329 (2004).13: 325330. [55] Sarma, Hiren Kumar Deva and Kar, Avijit. "Security Threats in Wireless Sensor Networks." Carnahan Conferences Security Technology, Proceedings 2006 40th Annual IEEE International. 2006, 243251. [56] Sen, Arunabha, Roxborough, Tom, and Sinha, Bhabani P. "On an Optimal Algorithm for Channel Assignmentin Cellular Networks." ICC'99 International conference on communication. 1999. [57] Sharma, Mohan B., Mandyam, Narasimha K., and lyengar, Sitharama S. "An Optimal Distributed DepthFirstSearch Algorithm." CSC '89: Proceedings of the 17th conference on ACM Annual Computer Science Conference. New York, NY, USA: ACM, 1989, 287294. [58] Sidek, Othman and Yahya, Abid. "Reed Solomon Coding for Frequency Hopping Spread Spectrum in Jamming Environment." Ammerican Journal of Applied Sciences. vol. 5. 2008, 12811284. [59] Sivakumar, Raghupathy, Das, Bevan, and Bharghavan, Vaduvur. "An Improved Spinebased Infrastructure for Routing in Ad Hoc Networks." Proc. of The Third IEEE Symposium on Computers and Communications (ISCC). 1998. [60] Strasser, Mario, Danev, Boris, and Capkun, Srdjan. "Detection of Reactive Jamming in Sensor Networks." ETH Zurich DINFK Technical Report (2009). [61] Strasser, Mario, P6pper, Christina, Capkun, Srdjan, and Cagalj, Mario. "JammingResistant Key Establishment using Uncoordinated Frequency Hopping." IEEE, 2008. [62] Sun, HungMin, Hsu, ShihPu, and Chen, ChienMing. "Mobile Jamming Attack and its Countermeasure in Wireless Sensor Networks." Advanced Information Networking and Applications Workshops, International Conference on 1 (2007): 457462. [63] Tague, Patrick, Slater, David, Poovendran, Radha, and Noubir, Guevara. "Linear Programming Models for Jamming Attacks on Network Traffic Flows." Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks and Workshops, 2008. WiOPT 2008. 6th International Symposium on. 2008, 207216. [64] Thai, My T, Tiwari, Ravi, and Du, DingZhu. "On Construction of Virtual Backbone in Wireless Ad Hoc Networks with Unidirectional Links." IEEE Transactions on Mobile Computing 7 (2008).9: 10981109. [65] Thai, My T., Tiwari, Ravi, and Du., DingZhu. "On Construction of Virtual Backbone in Wireless Ad Hoc Networks with Unidirectional Links." IEEE Transactions on Mobile Computing 7 (2008): 10981109. [66] Thai, My T., Wang, Feng, Liu, Dan, Zhu, Shiwei, and Du., DingZhu. "Connected Dominating Sets in Wireless Networks with Different Transmission Ranges." IEEE Transactions on Mobile Computing 6 (2007).7: 721730. [67] Thai, My T., Wang, Feng, Liu, Dan, Zhu, Shiwei, and Du, DingZhu. "Connected Dominating Sets in Wireless Networks with Different Transmission Ranges." IEEE Transactions on Mobile Computing 6 (2007).7: 721730. [68] Thai, My T., Zhang, Ning, Tiwari, Ravi, and Xu, Xiaochun. "On Approximation Algorithms of kconnected mdominating Sets in Disk Graphs." Theor. Comput. Sci. 385 (2007).13: 4959. [69] Thuente, David, Newlin, Ben, and Acharya, Mithun. "Jamming Vulnerabilities of 802.11 e." Proceedings of the 26th IEEE Communications Society Military Communications Conference (MILCOM). 2007. [70] Torrieri, Don J. "Fundamental Limitations on Repeater Jamming of FrequencyHopping Communications." Selected Areas in Communications, IEEE Journal on 7 (1989).4: 569575. [71] Toshihiro, Fujito. "On Approximability of the Independent Connected Edge Dominating Set Problems." FST TCS 2000: Proceedings of the 20th Confer ence on Foundations of Software Technology and Theoretical Computer Science. London, UK: Springer Verlag, 2000, 117126. [72] Toshihiro., Fujito. "How to Trim an MST: A 2Approximation Algorithm for Minimum Cost Tree Cover." ICALP 1 (2006): 431442. [73] Tseng, YuChee, Ni, SzeYao, Chen, YuhShyan, and Sheu, JangPing. "The Broadcast Storm Problem in a Mobile Ad Hoc Network." Wirel. Netw. 8 (2002).2/3: 153167. 142 [74] Wan, PengJun, Alzoubi, M. Khaled M., and Frieder, Ophir. "Distributed Construction of Connected Dominating Set in Wireless Adhoc Networks." Mob. Netw. Appl. 9 (2004).2: 141149. [75] Wang, Feng, Thai, My T, and Du, DingZhu. "On the Construction of 2connected Virtual Backbone in Wireless Networks." Trans. Wireless. Comm. 8 (2009).3: 12301237. [76] Wang, Hang, Guo, Jingbo, and Wang, Zanji. "Feasibility Assessment of Repeater Jamming Technique for DSSS." Wireless Communications and Networking Conference, 2007. WCNC 2007. IEEE (2007): 23222327. [77] Wood, Anthony D., Stankovic, John A., and Son, Sang H. "JAM: a JammedArea Mapping Service for Sensor Networks." RealTime Systems Symposium, 2003. RTSS 2003. 24th IEEE. 2003, 286297. [78] Wood, Anthony D., Stankovic, John A., and Zhou, Gang. "DEEJAM: Defeating EnergyEfficient Jamming in IEEE 802.15.4based Wireless Networks." Sensor, Mesh and Ad Hoc Communications and Networks, 2007. SECON '07. 4th Annual IEEE Communications Society Conference on. 2007, 6069. [79] Wu, Jie and Li, Hailan. "On Calculating Connected Dominating Set for Efficient Routing in Ad Hoc Wireless Networks." DIALM '99: Proceedings of the 3rd international workshop on Discrete algorithms and methods for mobile computing and communications. New York, NY, USA: ACM, 1999, 714. [80] Wu., Yiwei and Li, Yingshu. "Construction Algorithms for kconnected mdominating Sets in Wireless Sensor Networks." MobiHoc '08: Proceedings of the 9th ACM international symposium on Mobile ad hoc networking and computing. New York, NY, USA: ACM, 2008, 8390. [81] Wu, Yiwei and Li, Yingshu. "Construction Algorithms for kconnected mdominating Sets in Wireless Sensor Networks." MobiHoc '08: Proceedings of the 9th ACM international symposium on Mobile ad hoc networking and computing. New York, NY, USA: ACM, 2008, 8390. [82] Wu, Yiwei, Wang, Feng, Thai, My T, and Li, Yingshu. "Constructin kConnected mDominating Sets In Wireless Sensor Networks." October, 2007. [83] Xu, Wenyuan. "Channel Surfing: Defending Wireless Sensor Networks from Interference." Information Processing in Sensor Networks, 2007. IPSN 2007. 6th International Symposium on. 2007, 499508. [84] Xu, Wenyuan, Ma, Ke, Trappe, Wade, and Zhang, Yanyong. "Jamming Sensor Networks: Attack and Defense Strategies." Network, IEEE 20 (2006).3: 4147. 143 [85] Xu, Wenyuan, Trappe, Wade, and Zhang, Yanyong. "Antijamming Timing Channels for Wireless Networks." WiSec '08: Proceedings of the first ACM conference on Wireless network security. New York, NY, USA: ACM, 2008, 203213. [86] Xu, Wenyuan, Trappe, Wade, Zhang, Yanyong, and Wood, Timothy. "The Feasibility of Launching and Detecting Jamming Attacks in Wireless Networks." MobiHoc '05: Proceedings of the 6th ACM international symposium on Mobile ad hoc networking and computing. New York, NY, USA: ACM Press, 2005, 4657. [87] Xu, Wenyuan, Wood, Timothy, Trappe, Wade, and Zhang, Yanyong. "Channel Surfing and Spatial Retreats: Defenses against Wireless Denial of Service." WiSe '04: Proceedings of the 3rd ACM workshop on Wireless security. New York, NY, USA: ACM Press, 2004, 8089. [88] Zhang, Ning, Shin, Incheol, Zou, Feng, Wu, Weili, and Thai, My T "Tradeoff Scheme for Fault Tolerant Connected Dominating Sets on Size and Diameter." FOWANC '08: Proceeding of the 1st ACM international workshop on Foundations of wireless ad hoc and sensor networking and computing. New York, NY, USA: ACM, 2008, 18. [89] Zhang, Yifeng and Dill, Jeffrey. "An Antijamming Algorithm using Wavelet Packet Modulated Spread Spectrum." Military Communications Conference Proceedings, 1999. MILCOM 1999. IEEE. vol. 2. 1999, 846850 vol.2. BIOGRAPHICAL SKETCH Incheol Shin was born in 1977, in Seoul, Republic of Korea. He received Bachelor of engineering degree at computer engineering in 2002 from Hansung University, Seoul, Republic of Korea. In 2006, he received his Master of Engineering degree from the Department of Computer and Information Science and Engineering at the University of Florida. His major research area is computer networks. 145 PAGE 2 2 PAGE 3 Mostofall,IwouldliketoacknowledgemychairadvisorDr.MyTraThai.FromthemomentIstartedtoworkwithher,shehasencourageme,guidedmethroughalltheresearches,andgavemeinvaluableadvices,suggestionsandsupportstopursuethisdegree.Iamheartilythankfultothemembersofmysupervisorycommittee,Dr.RandyY.C.Chow,Dr.TamerKahveci,Dr.PrabhatMishraandDr.PanosM.Pardalos,fortheirguidanceandmentoring.Finally,Iwouldliketoshowmygratitudetomyfamilymembers. 3 PAGE 4 page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1ReactiveJammingAttacksinWSNs ..................... 12 1.2IdenticationofTriggerNodes ......................... 13 1.3ConstructionofRoutingBackbone ...................... 14 1.4Organization .................................. 15 2JAMMINGATTACKS ................................. 16 2.1EffectivenessofJammers ........................... 16 2.1.1PacketSendRatio(PSR) ....................... 16 2.1.2PacketDeliveryRatio(PDR) ...................... 18 2.2JammingAttackModels ............................ 19 2.2.1ConstantJammer ............................ 19 2.2.2RandomJammer ............................ 21 2.2.3DeceptiveJammer ........................... 22 2.2.4ReactiveJammer ............................ 23 2.3ExistingSolutions ............................... 25 2.3.1PhysicalLayerApproaches ...................... 25 2.3.2LinkLayerApproaches ......................... 31 2.3.3NetworkLayerApproaches ...................... 35 2.4Conclusion ................................... 36 3CENTRALIZEDIDENTIFICATIONOFTRIGGERNODES ............ 37 3.1NetworkModelandProblemDenition .................... 37 3.2Preliminaries .................................. 39 3.2.1MaximumCliqueProblem ....................... 39 3.2.2NonAdaptiveGroupTesting ...................... 39 3.3CentralizedTriggerNodeIdentication(CTNI) ................ 40 3.3.1GroupVictimNodesBasedonMinimumCollectionofDisjointDiskCovers(GVNMCDDC)Algorithm ................... 41 3.3.2DetectionofTriggerNodesBasedonNonAdaptiveCombinatorialGroupTesting(DTNNCGT)Algorithm ................ 41 3.4TheoreticalAnalysis .............................. 45 4 PAGE 5 45 3.4.2Correctness ............................... 47 3.4.3PerformanceAnalysis ......................... 48 3.4.4RandomReactiveJammingModel .................. 50 3.5TheTNLTCDSRoutingAlgorithm ...................... 51 3.6PerformanceEvaluation ............................ 52 3.6.1SimulationSetup ............................ 52 3.6.2ResultsandAnalysis .......................... 55 3.6.2.1PerformancebythenumberofjammersJ 55 3.6.2.2Performancebythenumberofradiosm 55 3.6.2.3PerformancebythenumberofnodesN 56 3.6.2.4Performancebythedensityofthenetwork ........ 56 3.6.2.5Performancebytransmissionrangeofthejammers ... 57 3.7Conclusion ................................... 57 4LOCALIZEDIDENTIFICATIONOFTRIGGERNODES .............. 59 4.1NetworkModelandProblemDenition .................... 59 4.2OverviewandFundamentalResults ..................... 60 4.2.1OverviewofIdenticationProcedure ................. 61 4.2.2HexagonTilingColoring ........................ 62 4.2.3Thek2ColoringAlgorithm ....................... 63 4.3LocalizedTriggerNodeIdentication(LTNI) ................. 65 4.3.1PartitionofNodesBasedonHexagonTilingandColoring ..... 65 4.3.2TriggerNodesDetectionProcedure .................. 68 4.3.2.1Sequentialgrouptestingbasedlocalizedtriggernodeidentication(SGTLTNI)algorithm ............. 68 4.3.2.2Identicationofasingletriggernode(ISTN)algorithm .. 69 4.4TheoreticalAnalysis .............................. 70 4.4.1UpperboundonTestingRounds .................... 70 4.4.2MessageComplexity .......................... 72 4.4.3RandomReactiveJammingModel .................. 73 4.5TheTNLTCDSRoutingAlgorithm ...................... 74 4.6PerformanceEvaluation ............................ 74 4.6.1TestingRoundsT 76 4.6.2Messagecomplexity .......................... 78 4.6.3Runtime ................................. 81 4.6.4Thenumberofnodesinquarantineareas .............. 83 4.6.5Randomreactivejammers ....................... 83 4.7Conclusion ................................... 84 5CONSTRUCTIONOFDOMINATINGTREE .................... 86 5.1OverviewofDominatingTree ......................... 86 5.2HardnessandApproximation ......................... 87 5.2.1Inapproximability ............................ 87 5 PAGE 6 .................... 89 5.3HeuristicAlgorithmandAnalysis ....................... 91 5.3.1AlgorithmDescription ......................... 92 5.3.2RuntimeComplexity .......................... 94 5.4PerformanceEvaluation ............................ 95 5.5Conclusion ................................... 99 6CONSTRUCTIONOFVIRTUALBACKBONEWITHMULTIPLEFACTORS .. 100 6.1OverviewofVirtualBackbone ......................... 100 6.2RelatedWork .................................. 102 6.2.1GeneralGraph ............................. 102 6.2.2UnitDiskGraph ............................. 103 6.2.3DiskGraphswithBidirectionalLinks ................. 104 6.2.4OtherResultsinCDS ......................... 104 6.3WirelessCommunicationModelandPreliminaries ............. 105 6.3.1Notations ................................ 105 6.3.2Terminologies .............................. 105 6.3.3Denitions ................................ 106 6.4MultiFactorsModelandSolutions ...................... 106 6.5ABetterAlgorithmforCDSonDiameter ................... 109 6.6ProgressiveAlgorithm(PA)andAnalysis ................... 113 6.7FurtherImprovementsforTheProgressiveAlgorithm ............ 119 6.7.1ReducingMultiplePaths ........................ 120 6.7.2RemovingRedundantTerminals ................... 121 6.7.3LocatingCentralArea ......................... 123 6.8PerformanceEvaluation ............................ 125 6.8.1PerformanceforCDSBDD,CDSBDandPA ............ 126 6.8.2PerformanceforBDAandPA ..................... 128 6.8.3PerformanceBasedonDifferent 130 6.8.4PerformanceforImprovementTechniques .............. 131 6.8.5PerformanceforMultiFactorsModel ................. 132 6.9Conclusions ................................... 133 7CONCLUSION .................................... 135 REFERENCES ....................................... 137 BIOGRAPHICALSKETCH ................................ 145 6 PAGE 7 Table page 31Notations ....................................... 38 61Runtime(ms) ..................................... 130 7 PAGE 8 Figure page 21Constantjammingattack .............................. 19 22Randomjammingattack ............................... 21 23Deceptivejammingattack .............................. 22 24Reactivejammingattack ............................... 23 31Sinceitem6(6thcolumn)isatriggernode(positiveitem),onlythe2ndand6thgroups(rows)returnnegativeoutcomes.Onthecontrary,allotherfourgroupsproducepositiveoutcomes. ........................ 44 325Possiblejammersactivatedbyatriggernodet 46 33Experimentalresultsbyvarioussizeofjammers ................. 52 34Experimentalresultsbyvarioussizeofchannels ................. 53 35Experimentalresultsbyvarioussizeofnodes ................... 53 36Experimentalresultsbyvariousnetworkdensities ................ 54 37Experimentalresultsbyvarioussizeof 54 41Theminimumdistancebetweentwonodeswithsamecolor ........... 64 42Thecoloringpatternfork=4 65 43Triggernodesinahexagon ............................. 71 44Roundsbyvariousparameters ........................... 76 45Messagesbyvariousparameters .......................... 77 46Runtimebyvariousparameters ........................... 78 47Nodesinquarantineareasbyvariousparameters ................ 79 48ThenumberofroundsTinrandomreactivejammingmodelwithdifferentvaluesofjammingprobabilityP. .............................. 80 51ReductionfromWDSGtoDTG0 89 52AnexampleofreductionfromGtoG' ....................... 90 53TheexecutionofHeurDTalgorithm ......................... 93 54SimulationresultsforHeurDT,MSTLandoptimalresults ............ 96 8 PAGE 9 ....................................... 106 62Anexampleforsecondphaseinalgorithm 12 ................... 112 63AllthenodesintheringareaCDSwithdiameterof8 .............. 117 64Anexampleforreducingmultiplepath ....................... 121 65Anexampleforremovingredundantterminals ................... 122 66Anexampleofleafnodeslocatedatcentralarea.Theblacknodesconsistofthetree. ........................................ 124 67AnexampleofselectingthenodewithmaximumdegreeastherootofCDS .. 125 68PerformanceforCDSBD,CDSBDDandPA ................... 127 69PerformanceforBDAandPA ............................ 128 610Performancebasedondifferent 130 611Performancefortherstandsecondimprovementtechniques .......... 131 612Performanceforthethirdimprovementtechniques ................ 132 613Performanceformultifactorsmodel ........................ 133 9 PAGE 10 10 PAGE 11 11 PAGE 12 8 ].Jammingattacks,oneoftheDoSattacks,especiallyarelightweighted,butthemostfatalthreatstoWSNs,becausetheyattackthecoreofwirelessbroadcastadvantageevenwithoutmodifyingcommunicationpacketsandcompromisingwirelesssensordevices.Duetotheexcessivehardwarerequirementonexistingmethods,threatscannotbeneutralizedandnulliedwithconventionalsecuritysolutions. 12 PAGE 13 78 ].Secondly,incasetriggernodesneedstosendmessages,they 13 PAGE 14 14 PAGE 15 2 describestheevolutionofjammingstrategiestoreactivejammersandexistingcountermeasureagainsttheattacks.Fortheidenticationphase,Chapter 3 providesthesolutionforcentralizedidenticationoftriggernodes,meanwhilethelocalizedalgorithmforthedetectionoftriggernodesispresentedinChapter 4 .OurproposednewproblemofdominatingtreestudiedinChapter 5 followedbythemultifactormodelinChapter 6 soastoprovidetheefcientconstructionofjammingresistantroutingprotocol.Chapter 7 concludesthisdissertation. 15 PAGE 16 52 55 69 84 86 ]describingjammingattacksbasedonthesemeasurements.Inthissection,theimportanceofjammingattacksandhowpowerfultheyarewillbeintroducedastheyapplytoPSRandPDR. IntendedToBeSentPacketsInmostMAClayersinWSNs,carriersensingmultipleaccess(CSMA)controlhastobeperformedbeforelegitimatecommunicationsinsendernodes.Thatis,senderdevicesarerequiredtosenseactivityinthechannelforacertaindurationoftimebefore 16 PAGE 17 17 PAGE 18 ReceivedPacketsACyclicRedundancyCheck(CRC)isasimplenonsecurehashfunctionfordetectionoferrorsincommunicationmessages.UsuallytheCRCcomputationwouldbedoneinthelinklayerbyalongdivisionoperationwherequotientsarediscardedandremaindersfromtheoperationbecomestheresultsoftheCRC,whichareappendedtotheendoflinklayerframes.ReceiverswouldbeabletodecodeandvalidatereceivedpacketswitheasebyutilizingtheCRCcheck.ThePDRratiodropsifjammersinjectinterferencesignalsintocommunicationmessages.Ajammerwouldbeabletodecreasethenumberofpacketsthataresuccessfullydeliveredtodestinationnodesbycorruptingongoingcommunicationpackets.Duetothebroadcastingnatureofwirelesscommunications,collisionsamongpacketsareconsideredacriticalproblem,andthejammersattackthePDRfromtheviewpointofthisweaknessbycorruptinginterferencesignals.AttackingthePDRnotonlydropsthePDR,butalsowastestheenergyofsensornetworksduetothefactthatcollidedmessagesrequireretransmission.Thesensornetworksareresourcelimitednetworks,andenergyconsumptionhasbeenmuchstudiedbecauseitdominatesthenetworklifetime.Inotherwords,invalidationofpacketsbyCRCcheckingimplyretransmissionsinsendernodesafternoticationsfromreceivers,whichwastessignicantlyenergyandresultsinashorternetworklifetime.Asaresult,wecansaythatattackingthePDRfromjammerscausesmoreseriousproblemsthanattackingthePSR.Inaddition,intermsofstealth,itwouldbebetterforjammerstodropPDR.ItwouldnotbenecessaryforjammersattackingPDRtoexposethemselvestolegitimatenodes 18 PAGE 19 Constantjammingattack AccordingtotheFig. 21 ,theconstantjammer,atrivialjammer,isthemostintuitivestrategytoimplement,butithastheleastenergyefciencyamongalljammerssinceitdoesnotfollowtheprotocolsinWSNs,justemittingaseriesofnoisesignalsintothe 19 PAGE 20 20 PAGE 21 Randomjammingattack 22 ,therandomjammerhasevolvedfromtheconstantjammertoconserveenergy.Liketheconstantjammer,itemitsjammingsignalsforacertainamountofthetime,butsleepsafterturningitsradiooffinordertoconserveenergyforalongerlifetimeofjamming.Whiletheswitchingmechanismbetweensleepingmodeandjammingmodecansaveasignicantamountofenergy,itwoulddroptheefciencyofattacktoPSR.Because,duringitsjammingphase,normalsenderswouldnotbeabletotransmitmessagesforchannelactivitiesunderjammingsignalsandwouldhavetowaituntilitssleepingphasebegins.Theycouldsendoutthestalledmessagesduringthejammingphase,andmostofthemessageswouldbesuccessfullydeliveredwithoutinterferenceduringthejammers'sleepingmode.Thisbecomesatradeoffbetween 21 PAGE 22 Deceptivejammingattack BymonitoringtheprotocolsinWSNs,thisisthersttypeofjammerthattakesthestealthofjammersintoconsideration.ThedeceptivejammerinFig. 23 doesnotsendoutrandombitsorwaveformsbyagenerator,butregularpacketsinordertocapturethechanneloflegitimatecommunicators.Thedeceptivejammersemittingregularpacketsforceslegitimatecommunicatorsintothereceivingstateandpreventsthemfromconvertingtheirstateintosendmode.Thismethodwasinitiallyimplementedbycontinuouslysendingoutpreamblemessages,sothatitishardtodetectandisaneffectivemethodtodisableaCSMA.ThedeceptivejammernotonlywoulddropthePSR,butalsoincreasethestealthagainstthedetectionsystembytransmittingregularframesintoaMAClayerchannel. 22 PAGE 23 2 ]alsomentionsaperiodicjammer,anotheradaptationofdeceptivejammer.Insteadofsendingoutcontinuouspreamblemessages,itgeneratesaseriesofshortpulsesineveryDIFSinterval(50nanosec),sonormalnodesndthenetworkalwaysbusy. Reactivejammingattack Reactivejammingattacksareconsideredoneofthemostintelligentjammingstrategiesduetotheirreactivebehavior.ThereactivejammingattackinFig. 24 is 23 PAGE 24 2 ]withknowledgeoftheprotocolsisoneoftheimplementationsofareactivejammeranditcanhaveseveraldifferentstrategiestocorruptmessages,includingcontrolmessagesanddatamessages.Importantly,itwouldbeabletodistinguishbetweencontrolmessagesanddatamessagesbasedonthelengthofeachcommunicationmessageortheintervalsbetweentheminWSNs.Accordingtothejammingstrategies,therecouldbefourtypesofintelligentjammingattacks,suchasCTScorruptionjamming,Acknowledgecorruptionjamming,datacorruptionjamming,andDIFSwaitjamming. 24 PAGE 25 7 26 31 32 35 37 43 47 61 70 78 83 85 87 ],butthehighcomputationaloverheadofthesemethodsbadlyreducestheeffectinresourcelimitednetworkenvironments,suchasWSNs.Forexample,inthechannelsurngmethodsfrom[ 7 35 37 47 78 83 85 87 ]andfrequencyhoppingmethodsfrom[ 26 31 32 36 43 58 61 70 ],thetransmissionfrequencyorchannelischangedtoarangewherethereisnointerferencefromtheadversaries.ThesestrategiesarenotquitesuitableforWSNs,especiallyinmultichannelWSNs,sincethesensorshavetoscanallthechannelstodetectthejammingattacksandhoptonewfrequenciesallthetime,eveninthemiddleofacommunication.Duetothefactthatmostofthesensornodeshaveahalfduplextransceiveronthem,scanningthechannelsduringtransmissioncausescommunicationstallstochecktheavailabilityofthecurrentchannel.Frequentcommunicationstallsresultinalongertransmissiondurationandmoreenergyconsumption.Consequently,thesemethodscannotavoidhighoverheadandresourceconsumption. 25 PAGE 26 26 31 32 36 43 58 61 70 ]andcodedivisionmultipleaccess(CDMA)[ 14 21 30 53 76 89 ],formsofdirectsequencespreadspectrumcommunication.Sincemostjammingattacksarecategorizedintophysicallayerattacks,thebeginningofthedefensemechanismsstartinthephysicallayer.Thissectionwillintroduceexistingphysicallayercountermeasuresagainstthejammingattacks.Thefrequencyhoppingtechniquewasdesignedasakindofsecuresolutionagainstjamming,eavesdropping,tempering,etc,inwirelesscommunication.JammingattacksareabletoattackWSNswithpartialbandnoiseatthebeginningstageofjammingattacks,andthefrequencyhoppingsolutionisutilizedtodefendagainstthejammerswithpartialbandnoiseatthattime.TheinitialoutlineoftheconventionalFHmethodistouseeachfrequencyslottotransmitpacketsthroughoneoforthogonalsignalsduringacertainperiodoftime,signalinginterval.Thatis,thetransmitterhopsbetweensafefrequenciesbasedonapredenedalgorithms.However,themostimportantissuetoexploitfrequencyhoppingtechniqueishowtosecretlyestablishswitchingsequencesbetweentwocommunicationpartiesinordertofoilathirdparty.Thatis,reducingtheprobabilityofinferenceisthekeyofthistechniqueagainstjammingattacks.ApresharedsecretcodeforFHisnotfeasibleinwirelesscommunicationnetworksduetothedynamicbehaviorofsensornodesandthescalabilityofthenetworks.SincethemainissuesofFHapproacharehowtoassignthehoppingsequencesforshiftingfrequenciesandhowtosynchronizethemamongthenodes,therehasbeenmuchresearchregardingthosecriticalissuesinFHmechanisms.Toremovetheburdenofdistributingsequences,[ 61 ]proposedtheuncoordinatedfrequencyhopping(UFH)techniqueforanantijammingpointtopointschemetoestablishasecretekeybetweentwocommunicationparties.Thispaperintroduced 26 PAGE 27 42 ]hasbeenintroducedtoachieveamorespectralefciencythananyotherexistingFHdesigns.Inthetraditionalmodels,FHtechniquesrequireamuchwiderspreadbandwidththantheyactuallyusetotransfermessages,whichmeansthespectralefciencyfromthetotalnumberofavailablecarrierfrequenciesistoolowinpracticalnetworkenvironments.[ 42 ]presentedaninnovativeformofMDFHthatexploitedmessagestreamsasapseudorandomsequenceforFHselection.Thedatastreamisdividedintomultipleblocks,andeachblockconsistsofadditionalcarrierbitvectorstodeterminethehoppingfrequenciesandordinarybitvectorsthatareactualdatatotransmit.AcarrierbitvectorhasBctospecifyahoppingfrequencytotransferthedataandNhthenumberofhopswithinasymbolperiod.Thedatablocksarefedintoaserialtoparallel(S/P)converterinordertosplittwocarrierbitsandordinarybitsintotwoparalleldatastreams.Inaddition,thispaperpresentedenhancedMDFHtoutilizemultipletransmissionsateachhopbyexploitingalltheavailablecarriers.Thissolutionhelpstoincreasetheresistanceagainstjammingattacksusinganunpredictablemessagedrivenhopping 27 PAGE 28 26 31 32 36 70 ],nocompletesolutiontotheproblemoffollowerjammingattackshassofarbeenreached.Thefrequencyhoppedmaryfrequencyshiftkeyed(FH/MFSK)wasdesignedasacountermeasureagainstapartialbandnoisejammer[ 36 ].TheimplementationofFH/MFSKutilizesMnonoverlappingfrequencysynthesizersineachtransmitterandreceivertohopeachMFSKsignalsincetheperformancefromconventionalFH/MFSKimplementationisgreatlyeffectedbythedeviationofasinglecarrier.However,thisapproachrequirescomplicatedhardwaretorealizeinapracticalsystem,whichisnotinfeasibleforresourcelimitedsensornetworks.[ 31 32 ]introducedadifferentapproachasacountermeasureevolvedfromFH/MFSKagainstfollowerjammers.Themultipleorthogonalsignalsduringacertaintimeperiod,signalinginterval,wouldbeemittedineachfrequencyslotthatisdividedfromtotalspreadbandwidth.Furthermore,thereisconventionalmodeandunconventionalmodeinordertosynchronizecommunicators.Inthecaseofconventionalmodefrompseudorandomprobabilityofpcinbothsendersandreceivers,areceiverwoulddeterminethe 28 PAGE 29 89 ]wasdesignedtobeimmunetoaninterferencesignal,butthisrequiressophisticatedmodulatinghardwareinthesensors.ThemainissueoftheDSSStechniqueisalsosecretkeysharingduetothefactthattheperformanceofsuppressioncapabilityislimitedbythepseudorandomgeneratorwiththekeysamongcommunicators.Thekeyestablishmenttodriveidenticalspreadingcodesamongthecommunicatorsisacriticalissueforthewirelessadhocnetworksintermsofscalability.ThesecretsfortheDSSScommunicationbetweenthesenderandthereceivershavetobeinagreementbeforethestartoflegitimatecommunications,butthispresharedsecretkeyhasbeenconsideredadifcultproblemtosolveduetothedynamicbehaviorofsensors.[ 53 ]proposedasolutioncalledUncoordinatedDSSS(UDSSS)forauthenticspreadspectrumantijammingcommunicationinordertosolvethekeysharingproblem.ThisapproachenablestheimplementationofDSSSwithoutpriorestablishmentofsecretkeysamongthecommunicators,andreceiverskeepacerticateofthesender'spublickeyinsteadofsharingsecretkeys. 29 PAGE 30 14 ]relatedtoutilizingthedirectsequenceCDMA(DSCDMA),resortstohighpowerdynamictreeremergingschemestomaintainthesmallnumberoforthogonalcodesinuseandtoavoidrecalculationofthecodes.SpreadspectrumcommunicationhasbeenstudiedtoresistjammingattacksinunicastcommunicationbecausethenumberofcodesgreatlyeffectstheperformanceofDSSS.[ 14 ]developedabroadcastingtechniquewithaDSSSschemewithfewernumberofcodesbyusingabinarykeytree.Duetothevariationinnodesindynamicenvironments,thismethodsuffersfromtheadditionalmaintenanceoverheadofjoininandleavingbehaviors,especially,socomputationoforthogonalcodestakesmuchtime.Inaddition,becauseoftheprobabilisticnatureofpacketreception,italsohasaproblemoffalsealarms,andasasolutionforfalsealarms,networksareperiodicallyrequiredtoresettheircodeineachsensorandbuildthecodetreerepeatedly.Thisseriesofreconstructionproceduresnecessitateshighmessagecomplexityandadditionalcomputationoverheadonthenetworksaswell.Intheserespects,aschemewithcryptographickeymanagementhasascalabilityproblemandastabilityproblemwhenappliedtovariousdynamicnetworks. 30 PAGE 31 30 ]aswellsincetherepeaterjammerstrytoacquirethecodebymonitoringtheongoingtrafcandgarblethecommunicationmessagesbasedontheacquisitionofthecode.[ 76 ]haveanalyzedDSSStechniquesattackedbyrepeaterjammersindetail,butsofartherehasbeennocompletesolutionforrepeaterjammingattack.Duetothenatureofajammingattack,mostdefenseschemesstartfromthephysicallayer,FHandDSSS.FHmethodsfrom[ 26 31 32 36 43 58 61 70 ]hopcommunicationfrequenciesseekingasafeoneinordertoavoidthejammedcommunicationsbasedontheswitchingsequences.Howtodistributethehoppingsequenceandsynchronizecommunicationpartnerswiththenewsequencesarethemainissuestoovercomelowspectralefciency.CodeDivisionMultipleAccess(CDMA)schemecommunication[ 14 21 30 53 76 89 ],aformofdirectsequencespreadspectrumisalsooneofthemostcommonwaytoresistjammingattacks.However,theproblemofhowtomanagesecretkeysforefcientsuppressioncapabilityhastobesolvedforbetterperformanceofimmunityagainstattackinresourcelimitednetworks. 7 35 37 47 78 83 85 87 ]regardingtheinvestigationofjammingattackfromtheviewpointoflinklayers.Inthissection,wedescribevarioustypesofevasiontechniquesinthelinklayer.EventhoughthechannelsurngmethodintheMAClayerwasmotivatedbyFH,itisthemostreactiveapproachbecauseitswitcheschannelsondemandaftervericationof 31 PAGE 32 83 ]introducedthechannelsurngstrategy,suchthatwhenthenodeswithdetectionsensorsarejammed,theyswitchtheircommunicationchannelintoanotherorthogonalchannelinordertoreconnecttotherestofanetwork.Theboundarynodesthatlosetheirneighborsfromajammingattackcandiscoverlostneighborsinnewchannelsandtrytorebuildtheconnectivityoftheentirenetwork.Therearetworeasonsnodeslosetheirneighbors,poorconnectivityandjammingattacks,andtheyprovidesimpleprotocoltoidentifythereasonforthelostneighborsbyanalyzingthechannelbeingusedforreconstructionbythelostneighbors.However,thisprotocolforcesnetworksintoanunstablestateduringconnectivityrebuildingduetofrequentlinkqualitydegradationsorthedynamicbehaviorofnetworks.Twomethodshavebeenproposedinordertorestorenetworkconnectivityafterattacks.Therstisthecoordinatedchannelswitchingtechniquewhenanentirenetworkswitchesitscurrentchanneltoanewchannelsoastoreconstructnetworkconnectivity.Thistechniquesuffersfromunreliablelinks,sothatsomenodesmightmissthenoticationtoshifttheirchanneltoanewone.Thesecondapproachisthespectralmultiplexingtechniquewhereboundarynodesactasbridgestoconnectthenodesofoldchanneltothenodesofnewchannel.Thisapproachenablesthenetworkstomaintainmultiplechannels,sothattheentirenetworkdoesnotneedtonotifyallnodes,justsome,toswitchtoanewchannel.Thereare,however,severalchallengingproblemstorealizingapracticalsystem,synchronization 32 PAGE 33 87 ]alsodescribedaspatialretrievalmethod,calledaphysicalevasionmethod,byphysicalrepositioningofmobilenodesoutofjammedregions,butthenetworkswouldbeunbalancedandevenisolatedbytheattacks.Sincetheyassumestationaryjammerswithmobilenodes,thenodeswithinajammedregionwouldbeabletoescapefromthejammedregionafterthepresenceofjammerswasdetected.Thechallengingissuesofthephysicalevasionmethodarehowtodeterminewhichdirectionsnodesshouldretreatandhowfartheyshouldretreatfromtheircurrentpositions,becausethesedecisionsmaycausedisruptionofnetworkconnectivity.Inaddition,thispapermentionedjammednodesmovingintoradiorangeafterrelocations,butthiswouldresultinshorternetworklifetimeaswell.Asanothertypeofevasiontechnique,[ 85 ]designedatimingchanneltorecoverreliablecommunicationlinksafterjamming.Thetimingchannelisalowratelayeroverphysical/linklayersusedtodetectthetimingofinterferedpacketsinthereceiversbyutilizingCRCcheckormonitoringsignalstrength.Thisapproachisforpointtopointcommunicationlinks,notforbroadcastcommunications.Thecriticaldependenciesofthetimingchannelschemearehowtodetecttheexacttimingofthefailurepacketreceptionsandhowtomaptheoccurrenceoffailedpacketstotheinformationtobedelivered.[ 7 ]proposedajammingresistantMACprotocolthatadjuststheprobabilityforsuccessfultransmissionbymonitoringchannelactivities,andeachnodewouldbeabletotransmitmessagesbasedonprobability.Theprotocolalsodividesthetimeintosmallerorbiggertimeintervalsaccordingtosuccessfulmessagetransmissionsinordertoadapttransmissiontime.Themainideabehindthisisthatadversariesobservetheactivityofthecurrentchanneland,ifthereisnotenoughactivity,theywouldnotheavilyjamthechannel.However,thisprotocolshouldincludeamechanismtodeterminesuccessfultransmissionsofmessagesinremotenodes,whichmeansthatother 33 PAGE 34 78 ]inordertoreducethedamagefromjammingattacksoncommunicationpackets.TheyintroducedseveralstrategiestodefendtheMAClayeraccordingtothetypeofattack.Forexample,framemaskingagainstaninterruptingjammingattack,channelhoppingagainstanactivityjammingattack,packetfragmentsagainstascanjammingattack,andaredundantencodingmethodagainstapulsejammingattack.FramemaskingisaDSSStechniqueusingsharedkeysbetweenwirelessnodes.Thepacketfragmentationmethodwouldbeusedtotransmitamessageinmultiplefragmentsduringajammer'schannelactivityscanning,andtheredundantencodingmethodisusefulforthereceivertorecovercorruptedmessagesduetojammingattacks.TheywanttocombineallthesetechniquesintoaMAClayerprotocoltodefeatjammingattacks.Thefundamentallimitationsoneachdefensemechanismareremained.Thatis,framemaskinghasaproblemofkeydistribution,andchannelsurnghasserioussynchronizationprobleminpracticalsystemasdescribebefore.Packetfragmentationmethodmightdivideapacketintotoosmallfragmentswithadditionalredundantencodingdataforrecovery,whichmakestheapproachunfeasibleinrealcommunicationsystems.Therehavebeentwomainapproachestojammingattacksinthelinklayer,channelsurngandmodicationoftheMACprotocol.Bothbelongtoacategorycalledevasionmethods,andutilizethejammers'scanningtimetotransmitlegitimatemessages.Channelsurngschemesfrom[ 83 85 87 ]arereactiveintermsofswitchingchannels,butsynchronizationisacriticalissuetoimplementinapracticalsystem.ModicationofMACprotocolschemesrequireadditionalcommunicationoverheadamongthe 34 PAGE 35 77 ].Inthissystem,thejammednodescooperativelymapajammedregion.Jammednodesthatarewithinajammedareatransmitmultipleblindmessagestoannouncetheirjammedstatustothemappingnodesthatarenotinthejammedarea,buthavejammedneighbors.Themappingnodescommunicatewithothermappingnodestoisolatethejammedareaandtoidentifybridgenodes.Thebridgenodesparticipateinrelayingmessagesaroundthejammedarea.Onedeciencytothisapproachwouldbethepossiblyunnecessarilylargejammedregionbuiltagainstthereactivejammingattack.Asaresult,partsofthenetworkmightbeisolated.Thisisbecausemanynodesintheexaggeratedlylargejammedregionmaystillbeabletotransmitwithoutactivatingthejammers,yettheyareisolatedandthemessagedeliveriesareinterrupted.Duringthemappingprocedureamongthemappingnodes,theprotocolrequiresanexcessivenumberofcommunicationmessagestobuildadetourroutearoundthejammedregion.Themultipletrafctopologiesfrom[ 62 ]couldbeusedtoevadethejammednodesunderattackfrommobilejammers.Themobilejammersinthispaperwouldbeabletoidentifythecriticalbroadcastingpathsinordertopreventdownlinknodesfromreceivinganymessages.Thenodescooperativelyconstructthemultiplepathsandselectapath 35 PAGE 36 63 ]designedalinearprogrammingmodelforaspecictypeofthejammingattack,butitfocusesmainlyonaowbasedattackwithoutconsideringofaprotocolbasedattackmodel.Unfortunately,thismightnotapplytogeneraljammingattacks.[ 16 ]investigateanefcientschedulingtechniqueforbroadcastingmessageswhenunderajammingattack.Thisapproachshowsgoodperformanceonlywhentherearepowerlimitationonjammers,whichmightnotbeapracticalassumptionsinceusuallythejammersaremuchstrongerthanthenormalnodesinWSNs. 36 PAGE 37 22 23 ]couplingwithminimumcollectionofdisjointdiskcoverbasedgrouping,thissolutioncanidentifyallthetriggernodeswithlowoverheadintermsofrunningtime,computationandmessagecomplexity.Thetheoreticalanalysisandexperimentalresultsshowthatoursolutionperformswellintermsoftimeandmessagecomplexities,whichprovidesagoodapproachtodefendreactivejammingattacks. 37 PAGE 38 31 Table31. Notations Meaning n ni nij U k H() PAGE 39 9 ].Sofar,thebestpolynomialtimeapproximationalgorithmforthemaximumcliqueproblemwasdevelopedbyBoppanaandHalldorsson[ 9 ],andachievedanapproximationratioofn(1o(1)).In[ 9 ],Hastadshowsthatthisisactuallythebestwecanachieveanditcannotbeapproximatedwithinafactorthatofn1forany>0.Therearesomeotherresultsintheliteratureconcerningtheapproximationofthemaximumcliqueproblemonarbitraryorspecialgraphs[ 9 10 29 ].Inthischapter,themaximumcliqueproblemisappliedtoobtaintheupperboundofthenumberoftriggernodesbasedonthenumberofreactivejammers.Sinceajammercanonlybeactivatedbythenodeswithinacertaindistance,wecanconstructaunitdiskgraphofallnodeswiththeradiustwicethedistancetoestimatetheupperboundofthenumberoftriggernodes. 22 23 ]methodsaretominimizethetestingperiodbysophisticatedlygroupingandtestingtheitemsinpoolssimultaneously,insteadofindividuallytestingthem.Thewayofgroupingisbasedona01matrixMtnwherethematrixrowsrepresentthetestinggroupandtheeachcolumnrefersanitem.M[i,j]=1impliesthatthejthitemparticipatesintheithtestinggroup,andthenumberoftestingisthenumberofrows.Theresultofeachgroupisrepresentedasanoutcome 39 PAGE 40 23 ],wheretheunionofanydcolumnsdoesnotcontainanyothercolumn.Basedonthepropertiesofddisjunctness,thedecodingalgorithmtoidentifythetriggersbasedonthetestingresultsbecomesverysimple.Wejustneedtoremovealltheitemsappearedinanynegativepoolsandtheremainingitemarepositive[ 23 ].Inthisway,onlyO(1)testingroundsandO(tn)decodingtimeareneeded.ToutilizeGTforourtriggerdetection,weneedtosolvethetwomostchallengingproblems:(1)Howtogroupthenodestoavoidinterferencebetweentheresultsamonggroupssoastotestthesegroupssimultaneously.(Anytwogroupsarecalledinterferencefreeifanyjammerstriggeredbyeithergroupcannotjamtheothergroup).(2)Howtoaccuratelyestimatethevalueofdwhichistheupperboundofthenumberoftriggernodes.Sinceddeterminesthenumberoftests,thetighterdis,thebettertimeandmessagecomplexitieswecanobtain. 60 ].Thenwetestthesevictimstoidentifythetriggernodesbycallingtwosubprocedures:1)WeusetheGVNMCDDCalgorithmGroupVictimNodesBasedonMinimumCollectionofDisjointDiskCovertogroupasmanyasvictimnodeswithoutinterferencewitheachotherineachcover.Eachcoverincludesasetofdisjointdiskswherethecenterofeachdiskwillactasatestoutcomecollector.Eachofthedisjointdiskscanbetestedsimultaneously.2)Forasetofvictimsineachdisjointdisk,weusetheDTNNCGT 40 PAGE 41 2 .BasedonthisobtainedvalueDij,DTNNCGTconstructs 41 PAGE 42 3 .ConsiderFigure 31 asanexamplewherewehavetwojammernodesJ1andJ2.Nodesv1,v2,...,v9andv15,v16,...,v25arethevictims,andm=3.Accordingtoour 42 PAGE 43 5: PAGE 44 Sinceitem6(6thcolumn)isatriggernode(positiveitem),onlythe2ndand6thgroups(rows)returnnegativeoutcomes.Onthecontrary,allotherfourgroupsproducepositiveoutcomes. algorithm,twodisjointdiskswillbefoundandtwogroupsG11=fv1,v2,...,v9gandG12=fv15,v16,...,v25gareconstructedaccordingly.Testingwillbeconductedonthesetwogroupssimultaneously.Forsimplication,Figure 31 justshowsthedetailtestingofG11.AftertheestimationofDij=1,ouralgorithmwillconstructa1disjunctmatrix.Basedonthismatrix,therstthreerowswilldoaonehopbroadcastmessagetothree 44 PAGE 45 31 wherethesecondandsixthrowshaveapositiveresult.Basedonasimpledecodingmethodmentionedearlier,wecaneasilydetectv6asatriggernode. 3.4.1EstimationofTriggerNodeUpperBoundDijInordertoconstructddisjunctmatrixfortestingsintestinggroupGij,weneedtoobtainanupperboundontriggernodes.Weassumethattheinterferenceradiusislargerthanlegitimatetransmissionradius,R=rwhere>1sincejammershavemorecapabilitiesthannormalsensornodes.LetJbethesetofjammersthattriggernodetcouldactivate.WenotethatthedistancesfromjammersinJtotareatmostrwhilethedistancebetweenanytwojammersmustbelargerthanR=r.Otherwisejammerswillinvokeeachotherandrunoutofenergy.Wehavethefollowinglemma: 32 ,wherem=jJj.WehaveOJi(=(O,Ji))r8i=1...mandJiJj>R=r81i PAGE 46 Followingthelemma,wehave: jJj1when2. jJj2whenp jJj3whenp jJj4whenq 2. jJj5when1. 5Possiblejammersactivatedbyatriggernodet PAGE 47 Proof. ,thenodesingroupGijcantriggeratmostjJijjjammers.Intuitively,weknowthatasetoftriggernodestoactivatethesamejammerhaveadistancelessthan2r.Inalgorithm 2 ,weconstructaunitdiskgraphGij=(Wij,Eij)withdiskradius2rsothatthenodeswhichtriggerthesamejammermustformacliqueingraphGij.Ineachiteration,accordingtoAlgorithm 2 ,wechoosetherstJthijmaximumcliquesandunionallthesecliques.Thatis,SJijk=1ck(Gij).Thustheproofiscomplete. Lemma2. Proof. Proof. 47 PAGE 48 Proof. 2 andthefactthateachdiskD1vhasaradius(Rr),allthevictimnodesinanytwodifferentdisjointdisksareinterferencefree.Thusthetestingresultiscorrectlycollected. Lemma4. Proof. 48 PAGE 49 Proof. 22 23 ].InWSNs,aswedenedtherearemradiossothatatmostmgroupscanbetestedatthesametime.AccordingtoTHEOREM ,dijareboundedbyDijandnijisthenumberofvictimnodes,wecompletetheproof. Proof. andCOROLLARY ,thecoversforallvictimnodesare(H)+1andthetestingtimeforeachcoveristhemaximumtestingtimeamongallgroups,thatis,maxjdminf(2+o(1))D2ijlog22nij PAGE 50 22 23 ]forddisjunctmatrix,hasthelowestupperboundforthematrixsize.Itistrivialtondthat,eachcolumnhasexactlys1entriesinthematrixconstructedinthatway,wheres=(2+o(1))Dijlog2nij Wedonotconsiderthefalsenegativefromrandomdelayonemittingadversarialsignalfromjammerssinceaccordingtothedenitionofreactivebehavior,thejammerswouldonlyemitinterferencesignalsduringthelegitimateactivitiesonchannels. 22 23 ].Using(d,e)disjunctmatrixhelpstocorrectatmosteerrorsinthetestingoutcomes,thuswearestillabletocorrectlyidentifyallthetriggers.Wewouldliketonotethatinordertousethementioned(d,e)disjunctmatrix,weneedtoestimatetheupperboundofe.Inpracticalnetworkenvironmentsunderrandomreactivejammingattacks,wecouldestimatethisboundbyanalyzingthePacketDelivery 50 PAGE 51 ReceivedPacketsThereactivejammerswouldbeabletodropPDReffectivelybythereactivestrategy,however,wecouldusethisratioagainstthemtoboundtheunreliabletestingoutcomesinthecaseofrandomreactivejammingbehaviorsbecausePr(1PDR)istheprobabilityofemittingadversarialsignalfromjammers,andPDRisalsooneofwellknownprobabilisticmethodstodeterminethepresenceofjammingattacksbyasimplecalculation.InordertoachievehighlyaccuratePDR,Strasseretal.[ 60 ]introducedabiterroridenticationtechniquetodifferentiatejammedpacketsfromerrorscausedbyweaksignal(e.g.,becauseoffastfadingorshadowing).Consequently,theupperboundoferrorsovertestscouldbederivedfrominvestigationofPDRandsentfromsensorstobasestationsoastoconstructerrortolerantdisjunctmatrix. 64 ]onG0.Itiseasytoseethatthe 51 PAGE 52 Experimentalresultsbyvarioussizeofjammers 52 PAGE 53 Experimentalresultsbyvarioussizeofchannels Experimentalresultsbyvarioussizeofnodes basestation,weassume=rinthissimulation,whilelargerwouldmakethissolutionmoreefcient.Wehaveintotalsixbenchmarksinthesimulationswithdifferentinputparameterteams.Ononehand,westudytheaveragenumberofdiskcoverscintheGVNMCDDCalgorithm,andthemaximumnodedegreetovalidatetheboundofcprovedinLEMMA .Ontheotherhand,weshowtheoveralltestlength(numberofroundsT)analyzedinTHEOREM3.Moreover,werecordthenumberofvictimnodesnandthetotalvolume PAGE 54 Experimentalresultsbyvariousnetworkdensities Experimentalresultsbyvarioussizeof 5.2.1 3.6 3.6 3.6 and 3.6 arethecorrespondingresultsandanalysis.Notethat 54 PAGE 55 3.6.2.1PerformancebythenumberofjammersJFigure 5.2.1 (a)and(b)explainourprotocolperformancebasedonthevariousnumbersofjammersJinthenetwork.Inthistest,wehaveN=1000nodeswithm=3radios,ona15001500networkeld,whereJ2[1,10]jammersarerandomlydeployed.OurprotocolemploysasophisticatedtechniquetoperformasmanyparalleltestingaspossibleasshowninAlgorithm3,thereforethenumberoftestingrounds,T,canbestablewhilethenumberofjammersJandvictimnodesnincrease.AsshowninFigure 5.2.1 (a)and(b),Tincreasesalittlewhilencanvaryfrom50to450whenJincreasesfrom1to10.Morespecically,thenumberofdiskcoverscandmaximumnumberoftestingroundsperdisktaresmallerthan10,wherethelatterismuchsmallerthanmaximumnodedegree.ThiscontributestodramaticallysmallnumberofoverallroundsT,whichisnolargerthan30andstableforincreasingJ.Moreover,sinceeach(Rr)diskinourtestsneedsonlyonesensornodetosendtheresultbacktothebasestation,themessagecomplexityMisalsomuchsmaller(lessthan100)thanthenumberofvictimnodesn.Notethatinindividualtestingmethod,MshouldbeashighasO(n).Therefore,oursolutioncanpromptlydefendajammingattackwithincreasingnumberofjammers,intermsoftimecomplexityandmessagecomplexity. 55 PAGE 56 3.6 (a),themaximumtestingroundsperdiskdecreaseastheradiosizeincrease,whichassiststodropthetotalrounds,T,drastically.Especially,whenm=2fromm=1intheFigure 3.6 (a),theoveralltotalroundsdroprapidly.Inconclusion,welearnthattheradiosizecanhighlybenettheoveralltestlengthofourprotocol. 3.6 (a)and(b).AsshowninFigure 3.6 (a)and(b),thevictimquantityincreasesobviouslyasthenumberofnodesincreases,butthenumberofmessagesisquiteconstant.Moreover,thetotaltestingroundsincreaseslowly.ThisgureshowshowoursystemefcientlyoperateswhenthenumberofnodesincreasesfromN=500toN=1000withm=3andJ=5jammersina15001500networkarea.Fromthisevaluation,wecanconcludethatourmodelisalsoaverysuitablesecuritysolutionforthemajorityofsensornetworksinvariousareas. 56 PAGE 57 3.6 (a)and(b)showsthevarioussimulationresultswiththeincreasingnetworkeldsizefrom15001500to25002500whereN=1000withm=3andJ=5jammers.Asthenetworkissparse,thenumberofvictimnodesdecreasesasgetssmallerinthisgureaswediscussed. 77 ]locksdownthewholejammedregionwhileoursystemminimizesthejammedregionsizebyidentifyingthesmallernumberoftriggernodes.InFigure 3.6 (a)and(b),asthegetslarger,thenumberofvictimnodesincreasessinceajammercantransmitfartherandcontaminatesmorenodesduringtheactivation.Moreover,morevictimnodesrequiresmoretestingroundstoculloutthetriggernodesamongthem.Inourresult,thenumberofroundsrisesasgetslarger.However,thenumberofroundsischangingveryslowly. 57 PAGE 58 22 23 ]discussedinSection 3.4.4 58 PAGE 59 59 PAGE 60 60 ]proposedahighlyaccuratedetectionschemeofreactivejammingattack,andallinterferencesignalscanbecorrectlyidentiedasnoisebysensorsfromotherexternalinterferenceseveninlowerandunsteadyRSS.However,individualtestingistootimeconsuming,thusweoftentestanumberofnodes,calledgrouptesting(GT).However,testingagroupofnodessimultaneouslyencounterseveraldifculties.Forexample,ifsomeNoisesensedafterperformingtesting,wemaynotknowwhichonesinthetestednodestriggeredthejammers.Moreover,schedulingnodesinatestinggrouptoperformthetestingsynchronouslymayresultinalotofcommunicationoverheadinthenetworkiftestednodesarefarfromeachother.Inaddition,iftwogroupofnodesaretestingatthesametimeandthejammertriggeredbytherstgroupcanjamnodesinthesecondgroup,the 60 PAGE 61 1. Iftwonodesu,vareatthedistanceatleastR+rtheycannottriggerasamejammer.Thisenableustotestu,vinasameroundwithouthavingtheoutcomeoftestinguandthatofvinterferedeachother.Ingeneral,wecanperformtestinginparallelfortwosetsofnodesUandVthatareR+rfarawayfromeachother. 2. Ifu,v,wareidentiedtriggers,thenallnodesinsidethetrianglewhoseverticesareu,v,warealsotriggers.Furthermore,ifT=ft1,...,tkgisasetoftriggers,thenallthenodesinsidetheconvexhullofTarealsotriggers.ThisholdsaslongasR>2r. 61 PAGE 62 4.3 tolocallypartitionsensornodesinagivenWSNintoasetoftestinggroups.Inordertostudythehexagontilingcoloring,weconsiderthefollowingnewproblem: Hexagontilingcoloringproblem:Givenadistanced2<+andahexagontilingHdividingthe2Dplaneintoregularhexagonsofsides1 2.FindtheminimumnumberofcolorsneededtocolorH,suchthatanytwohexagonsh1andh2inHwithsamecolorareatdistancegreaterthand.Thedistancebetweentwohexagonsh1andh2,denotedasd(h1,h2),isdenedastheEuclideandistancebetweenanytwoclosestpointsp1andp2,suchthatp1islocatedinh1andp2islocatedinh2.Thismakesthehexagontilingcoloringproblemdifferentfromthechannelassignmentproblem[ 56 ]incellularnetwork,wherethedistancebetweentwohexagoncellsismeasuredfromtheircenters.centersofallthehexagonsareplacedonatriangularlattice.Therefore,weconsideranewcoordinatesysteminthe2Dplane,withaxesinclinedat60o.Thisnewcoordinatessystemhastwounitsvectors!i(p 2,0)and!j(p 4,3 4)asshowninFigure 41 .Thecentersofeachhexagonhcoincidewiththeintegralcoordinatesinthis 62 PAGE 63 2p 42 showsthecoloringpatterngeneratedbythek2coloringalgorithmford=3p 2andk=4.Thek2coloringalgorithmisusedbythesensornodesinourproposedlocalizedalgorithmtolocallyidentifythegrouptheybelongto. Lemma6. Proof. 2(k1)(wherek=l2d PAGE 64 Theminimumdistancebetweentwonodeswithsamecolor Letx=i1i2andy=j1j2.Itfollowsthatxandywillbemultipleofk.Thedistancebetweenthecentersofh1(i1,j1)andh2(i2,j2)isgivenbydc(h1,h2)=p 2p 4(x 16x2q 16(2k)2>p 2(k1)+1.Notethatforeveryhexagonthedistancefromapointinsideittoitscenterisatmost1 2.Hence,thedistancebetweentwohexagonsd(h1,h2)willbeatleastdc(h1,h2)2(1 2)>p 2(k1). 2)>q 4k21>p 2(k1).Otherwisethereareonlysixleftcasesofx,yasshowninFigure 41 .Thedistancebetweentwohexagonsinallofthesecasesisexactlyp 2(k1).Hence,thelemmaiscompletelyproved. 64 PAGE 65 Thecoloringpatternfork=4 65 PAGE 66 4 ,anodecanidentifythecolorofitshexagonandalsothetimeslotassignedtoitsgroupintheschedule.Weconsiderthatanodev2VknowsitsneighborsN(v)andusingsomeadhocpositioningmethod[ 48 49 ],itcanidentifyitslocationas(xv,yv)withrespecttosomereferencenode.Weconsiderthesinknodes2VintheWSNasthereferencenodesuchthat(xs,ys)=(0,0).Now,weshowthatifanodevknowsitscoordinates(xv,yv)intheCartesiansystemthenwithouthavingtheglobalviewofthehexagontiling,itcanlocallycomputeitscoordinates(xhv,yhv)innewcoordinatesystemonthehexagontilingandfurther,itcanidentifythehexagonitbelongsto.Forinstanceanodevatcoordinates(xv,yv)intheCartesiancoordinatescancomputeitscoordinates(xhv,yhv)inthenewcoordinatesystemas: Thecoordinatesofthehexagonh(i,j)inwhichnodevislocatedisgivenas: 2% 2% Now,usingthek2coloringalgorithmandconsideringd=R+r,k=l(R+r)=p 2m=l2(+1) Inordertoshowthecorrectnessofourmethod,weprovethefollowinglemmas: 66 PAGE 67 Proof. Proof. 4 tocolortheentirehexagontilingis: (2)+1)m2,whenDh=R2r. Proof. Dhp (2)+1)m2colors 67 PAGE 68 7 ,basedontheconstraintondistancebetweenanytwojammersj1andj2,d(j1,j2)>Rinordertoavoidmutualinvocationbetweenthem,weprovedthatonlyonejammercanbeactivatedbynodeswithinahexagon.ReactiveJammingModel 4.3.2.1Sequentialgrouptestingbasedlocalizedtriggernodeidentication(SGTLTNI)algorithmInSGTLTNIalgorithm,byusingSGT,alltriggernodescanbeidentiedinO(jCTjlog)whereCTistheconvexhullofthesetoftriggernodeswithinahexagon(seeFigure 43 )andisthemaximumdegreeofallnodesinthenetwork,hence,therearenomorethan+1nodeswithinahexagon.WeuseamethodnamedQuickIdenticationinordertoreducethenumberoftestingrounds.AccordingtoLemma 7 ,alltriggerswithinahexagonactivateasamejammer.Thus,anodeisatriggeriffitbelongstotheintersectionofthehexagonwiththediskofradiusrwhosecenteristhejammer.Theconvexityoftheintersectionarea 68 PAGE 69 5 .Ineverystep,Tdenotesthesetofidentiedtriggers,Udenotesthesetofunidentiednodes.WeuseISTNalgorithm(presentedshortlyafter)tondamongUasingletriggervtthathasthemaximumdistancetothetemporaryconvexhullofT.Weshowlaterthatvtmustbelongtothethe(nal)convexhullCTofalltriggersinsidethehexagon.WesafelyeliminateallnodeswhosedistancesfromtheconvexhullofTarelargerthanthatofvt.WealsouseQuickIdenticationtoincludealltriggersinsidethenewconvexhullofT[fvtg.Thealgorithmterminateswhenallnodeswithineachhexagonareclassiedintoeithertriggersornontriggers. 6 .Thealgorithmworksinasamemannerwithbinarysearchalgorithmasitsequentiallydividesthesetintotwohalves.However,italwaystestsforthepresenceofthetriggersintherighthalfrstsothatifthereexisttriggersamongU,theonewiththemaximumindex(therightmosttrigger)willbereturned.ISTNterminatesassoonasonetriggernodeisidentied.Clearly,eachidenticationofatriggernodeamongasetUofnodesbyISTNtakesatmostlog2(jUj)rounds.RandomReactiveJammingModel:Incaseofjammerrandomlyreactswithprobabilityp,weproposeasimpleandeffectivealgorithmtoidentifytriggersinAlgorithm 7 .Asetofnodesareidentiedasnontriggersonlyifafterftestingrounds,noNoiseissensed.Werstrevealiftherearetriggerswithinthehexagonasinlines5to15.Ifitisthecase,furtherindividualtestsareperformedtoidentifywhethernodesaretriggers. 69 PAGE 70 sthen 4.4.1UpperboundonTestingRounds Lemma10. Proof. 5 requiresnomorethanO(jCTjlog)numberofroundswherejCTjisthenumberofverticesontheconvexhullofthesetoftriggers. PAGE 71 2(low+high)c stimethen Triggernodesinahexagon 71 PAGE 72 sthen sthen 9 andmaxfctgisthemaximumsizeofconvexhullofasetoftriggerswithinahexagon.Althoughmaxfctgmaygoupto,thealgorithm'sperformanceisoftenfarbetterthanitsworstcase. Theorem4.2. 5 isO(T)whereTisthenumberoftestinground. PAGE 73 5 ,thenumberoftestingmessageswithinahexagonaltestinggroupcannotbemorethanineachtestinground.TheobservationunderliesonthefactthatthemaximumdiameterofahexagonisatmostrfromtheSection 4.3 .Consideringthecasethatalltriggernodesconsistofconvexhullwithinahexagon,eachiterationoftestingroundwillidentifyatleastatriggernode.Hence,themessagecomplexityofalgorithm 5 isatmostO(T). 7 is1(1p)f.Theprobabilitythatatriggernodesiscorrectlyidentiedisalso1(1p)f(lines17to25).Hence,theexpectednumberofidentiedtriggerswillbe(1(1p)f)(1(1p)f)d>(12(1p)f)d.Toobtainanexpectedfalsenegativeratei.e.thefractionoftriggersthatareincorrectlyclassiedasnontriggers,weneedtosetf=dlog1p=2e.Forexample,ifp=3=4andthedesiredfalsenegativerate=0.01i.e.1%,weneedf=4.Notethat 73 PAGE 74 7 alsoworksforReactiveJammingModel.Simply,settingp=1,f=1wehaveanalgorithmwiththenumberofroundsisatmostc(+2).Clearly,thenumberofroundsdoesnotdependsonthesizeofnetwork(thenumberofsensornodes)butontheratio=R=randthemaximumdegreethatisoftendecidedbythedensityofthenetwork.Hence,theproposedalgorithmisscalablefornetworksofarbitrarysize. 77 ]approachintermsoflatency,messageoverheadandsizeofjammedregions(quarantinearea)aswell.Thepurposeofthesesimulationsistovalidateourapproachinvariousnetworkenvironmentsviadifferentteamofparametersinnetworkdensity,quantityofjammers 74 PAGE 75 r=3.Toinvestigatetheperformanceofourscheme,thenodesarevariedfromn=1000ton=5000inordertobesubjecteddensityofnetwork,atmostJ=10jammersareplacedforvarioussizeofjammers,andthetransmissionrangeofajammerisvariedfrom=R r=2.5to=R r=8.Experimentalimplementationofthesesimulationsdonotconsiderpacketlosses,linkcongestionorMACmisbehaviorexceptjammingsignalsinordertoevaluatetheidenticationperformanceonly.Sinceweranthesimulationsforeachsetup100timesandaveragedtheresults,theresultssufcetoreecttheefciencyofLTNIalgorithms.Wemodeledthepracticalnetworkstovalidateouralgorithmagainstreactivejammers,oneimplementationofthereactivejammers,byutilizingZigBeeprotocol.AwiderangeofexperimentswasconductedbasedonsimpleZigBeeprotocolusingCarrierSenseMultipleAccess/CollisionAvoidance(CSMA/CA)channelaccessmechanism.RequestToSend(RTS)ofsize30bytesandClearToSend(CTS)ofsize30bytesareimplementedintheseexperiments.Theprocessingtimeforanytypeofmessagesisuniformly10mssincesensorshavelimitedresourcetogeneratecommunicationmessages,andthepropagationspeedis3108m/sinbothalgorithms,JAMandLTNI.Tocommunicatewithothers,everynodeneedstosendatleastthreemessages,apairofRTS/CTSandadatamessageandwaitthepredenedintervalsbetweenthosemessages,20msintheseexperimentsinordertosimulatepracticalWSNs.Wealsoassumethatthesizeofacommunicationmessageisbiggerthan2347bytes,sothatRTS/CTSshouldbesentbeforelegitimatecommunicationsbegin.ByimplementingarealprotocolZigBee,wecouldreportmorereliableevaluationofouralgorithmbasedonthat. 75 PAGE 76 Roundsbyvariousparameters 4.6 (a),whichdirectlyreectsthelatencyofSGTLTNIalgorithmsduetothepredenedlengthofatestinground.AsshowninFigure 4.6 (a),thetestingroundsrequiredtocompletetheidenticationofalltriggernodesgrowsteadily,comparedtotheincrementalsizeofsensorsindensernetworks.Duringthenodesinincrementsfrom1000until5000,thetestingroundsgraduallyascendonlyaround120additionalroundsinSGTLTNI,butITLTNItestsatmost150additionalroundstodetectalltriggernodes.Thatis,thedesignofSGTLTNIalgorithmwithQuick 76 PAGE 77 Messagesbyvariousparameters IdenticationandQuickEliminationinordertoidentifyalltriggernodesonlyonconvexhullofeachhexagonproducesagreatbenetoverthetimecomplexity.Then,theimpacttoLTNIalgorithmsfromdifferentquantityofjammersaredepictedthroughFigure 4.6 (b)soastoshowtheeffectivenessofourlocalizedschemeinmassivejammingdisruption.Sincer=20and=R r=3withJ2[1,10]aresufcientconditionstoinvestigatemassivejammingattackovern=3000inWSNs,noadditionaljammersbeyondJ=10issimulated.Asjammersswellupto10timesofinitialsize,just130supplementarytestingroundstakeplaceinSGTLTNI.Consequently,evenwithmassiveimpactscenariofromlargenumberofjammersagainstWSNs,thelocalizedidenticationalgorithmproducesgreatrobustnessandfeasibilityonpracticalsystems. 77 PAGE 78 Runtimebyvariousparameters Finally,thetestingroundsshowsdiversityduetothedistanceamonginterferencefreetestinggroupsaccordingtothesizeof=R r,whileparametervaluebiggerthan8wouldbeimpracticalscenarios.AsindicatedbyFigure 41 ,disjointinterferencefreetestinggroupshavetobefarawayatleastR+r,thereforethedistancebetweenparalleltestinggroupsistightlyrelatedtothenumberofcolors.Duetothefactthatbiggerresultsinmorecolorswithsmallernumberofinterferencefreetestinggroups,Figure 4.6 (c)disclosesincreasingtrendsoftestsinbothLTNIalgorithms. 78 PAGE 79 Nodesinquarantineareasbyvariousparameters resultinlongernetworklifetimegenerally.ThegraphsinFigure 4.6 plotthenumbersofmessagespernodefromthreesolutions,JAM,ITLTNIandSGTLTNIsoastoreportcomparativemessagecomplexity.Figure 4.6 (a)providestheperformancecomparisonofthreeapproachesintermsofmessagespernodewhenchangingthenetworkdensity.ThemessagesfromJAMsolutionarehigherthanthosefrombothLTNIschemes.Inaddition,Figure 4.6 (a)showsthesuperiorityofmessagecomplexityinLTNIapproachesoverthatinJAMincreasesasmorenodesareplacedinnetworks.Thegraphexplainsthat40%moremessagesarerequiredtoconstructjammedareasinJAMwheren=1000,however,around47%lessmessagessufcetocompletetheclassicationofalltrigger 79 PAGE 80 ThenumberofroundsTinrandomreactivejammingmodelwithdifferentvaluesofjammingprobabilityP. nodesfromSGTLTNIintotalwheren=5000.Apartfromthenumberofmessagesbetweentwoapproaches,JAMsolutiondemandsinevitablymoreenergythanSGTLTNIdoessinceJAMprotocolhasacoupleofdifferenttypesofcommunicationmessagesincludingBUILDandPROBEmessagesexceptblindmessageJAMMEDinordertoquarantinejammedregions,butSGTLTNInecessitatesmainlysmallsizeoftestingmessagestoinvokejammersduringidenticationofalltriggernodes.Insummary,ourlocalizedalgorithmispromisingapproachtoapplyinpracticalnetworkswithaffordablemessagecomplexityandenergyconsumption.Simulationswerealsocarriedouttocomparetheperformanceofallthreeapproachesduringincreasingthenumberofattackers(jammers)aswellastoseehowthisaffectstheirperformances.AsrevealedinFigure 4.6 (b),bothLTNIalgorithmssignicantlyoutperformJAMapproach.Especially,thenumberofmessagesfromLTNIalgorithmsis32percentlessthanthatofJAMwhereasinglejammerJ=1,andLTNIschemesrequire45percentlessthanJAMdoeswhereJ=10.Inaddition,Whensizeofjammersisgreaterthan7,thenumbersofmessagespernodesobtainedfromthreeapproacheskeepconstanttrend,sincethejammerscoveredmostofnodesinnetworks. 80 PAGE 81 4.6 (c)showsthemessagespernoderequiredtocompletethreealgorithmsandnotonlymoremessagesinJAMthaninLTNIalgorithmsbutalsogrowingsizeofmessagespernodesinJAMapproach.Consequently,eveninseverewidespreadreactivejammingagainstWSNs,ouridenticationapproachvalidatesgreatrobustnessandpracticality. 81 PAGE 82 4.2.2 ,theoverallruntimeforcompleteidenticationofalltriggernodesdoesnotdrasticallygrowwiththesizeofjammersornodes.InFigure 4.6 (a),duringtheincrementsofnodesfromn=1000ton=5000,ourapproachdelayedonly2secondstoachievethetotalidenticationoftriggernodes,whereas,JAMtakes7timesaslongtimeasLTNIalgorithmsdo.Thatis,ouridenticationschemesintroduceoutperformanceonscalabilityovertheJAM.Asexpected,theruntimesobtainedfrombothLTNIalgorithmsshowssimilarincreasingtrendofcurves.Figure 4.6 (b)comparestheruntimebasedonvarioussizeofjammerswithxednumberofnodes,andSGTLTNIisthebest.Particularly,theruntimeofSGTLTNIisslightlybetterthanITLTNI,andthegapbetweenJAMandLTNIapproachesgetbiggerwhenjammersincrease.Asdescribedbefore,bigger=R rwithxednumberofjammersJ=3resultsinsmallernumberofinterferencefreetestinggroups,whichimplieslessparallelismonidenticationconsequently.Figure 4.6 (c)plotsthetimeofquarantineprocedures,andduetothebiggerimpactofjammersfrombigger,JAMalgorithmrequiresconsiderablylongertimetoquarantinejammedregionswhenincreasing.Inparticular,SGTLTNIandITLTNIshowsincreasingtrendofcurvesasgrowing.ConsideringtheruntimeofJAMapproach,thisvericationtimeofidenticationtriggernodesinLTNIalgorithmsisquitereasonable.Forexample,JAMdemands30secondstoblockjammedregions,butSGTLTNIcompletestheidenticationoftriggernodeswithinonly15secondswhere=8.Inaddition,ITLTNIalsoshowsgoodperformance,buthigherthanSGTLTNIdoes.Consequently,overalllengthofruntimedependsonthesizeofnodesinjammedregions,however,hexagontilingcoloringschemehelpstokeepsmallincrementsofruntimebymaximizingparalleltestings. 82 PAGE 83 77 ],andunreachabletriggernodesalsocannotreceiveanymessageeither.InFigure 4.6 ,thesizeofunreachabletriggernodesisalwayssubstantiallylessthenthesizeofjammednodesfromJAMalgorithm.Especially,inFigure 4.6 (a),onlylessthancoupleofnodesareunreachabletriggernodesandwouldnotbeabletoreceivemessageswheren2[1000,5000],butjammednodesgetsignicantlylargerashighernetworkdensity.InFigure 4.6 (b),aspredicted,thenumberofunreachabletriggernodeskeepsmallnumber,lessthan10,evenin10jammers,butjammednodessproutwithhigherpopulationfrommorejammersinWSNs.Withxednumberofjammersandsensornodes,largersizeofresultsinbiggerimpactagainstWSNs,whichimpliesthatmorejammednodesinJAMalgorithm.Yet,importantly,ouridenticationapproacheswillnotgetaffectedfromintermsofunreachabletriggernodes.Thatis,byutilizingthesuccessfulidenticationofalltriggernodes,actualjammedareasinwhichnonodewouldbeabletosendoutanymessagestoavoidreactivejammingsignalwouldbeverysmall,sothatsignicantlymorenodeswouldbeparticipatedinsecurecommunicationsthanJAMalgorithmdoes. 83 PAGE 84 4.6 thefalsenegativerateoftheAlgorithm 7 inalogscale(Recallthatwedonothavefalsepositive).Thefalsenegativeratelinearlydecreasesinthelogscalei.e.exponentiallydecreaseswhenthenumberofduplicatedtestingfincreases.Evenwiththep=0.6,thefalsenegativeratequicklydecreasesto1%withf=6.Thatis,weonlyneedtorepeatthetestfor6times.Whenwexthetargetedfalsenegativeratetobe1%,thenumberofroundsrequiredisshowninFigure 4.6 .Thenumberofroundsincreasesslightlytogetherwiththenumberofnodesinthenetworksinceputtingmorenodesinasamexedarearisethedensityi.e.degreeofnodes.However,therateofincreasecomesclosertozerowhenn=5000. 84 PAGE 85 85 PAGE 86 66 ].However,thisproblemonlyconsidersthesizeofthesetobtained,nottheweightofedges.Inotherwords,theweightisonlyassociatedwith 86 PAGE 87 6 72 ]studyingthetreecoverproblemwhichisdenedasaconnectededgedominatingsetwithtotalminimumedgeweights.Arkinetal.rstsolvedthisproblemin[ 6 ].Later,in[ 71 ],thisproblemcanbeapproximatedwithinafactorof3+.In[ 72 ],theauthorpresentedafastcombinatorial2approximationalgorithmforthetreecoverproblem.Incontrasttotheabovework,DTisdenedasnodedominatingsets,notedgedominatingsets.Usually,DTalwaysproducesasmallernumberoflinksandweightthantreecover,anditismoredifcultthantreecoverproblem. Theorem5.1. 25 ],WDSisshowntobeinapproximablewithin(1)lnjVjforany0,unlessNPDTIME(jVjloglogjVj).GivenaninstanceofWDS,thatis,agraphG=(V,E,w)andaweightfunctionw:V!Z+,theproblemistondaminimumweightsubsetofnodesthatdominatesallothernodes.WewillconstructaDTinstanceG'=(V',E',w')asfollows.Foreachnodevi2V,weintroducetwonewnodesxiandyishownasgraynodesandblacknodesinFigure 51B .LetX=fxigandY=fyig.LetV'=V[X[Y.ThesetE'andw' 87 PAGE 88 88 PAGE 89 ReductionfromWDSGtoDTG0 89 PAGE 90 12 ]andthuswecanobtainanDTofG.Itisclearthatthereductioniscompletedinlineartime.ThisreductionisshownasanexampleinFigure 52 AnexampleofreductionfromGtoG' 90 PAGE 91 12 ],theauthorsprovidedapolylogarithmicapproximationfortheDSTprobleminquasipolynomialtime.ThenwehaveanalgorithmtoobtainthisratiofortheDTproblemaswell: 12 ]inG'andmakevastherootrtogetaDT Algorithm 8 isbasedontheideaoftransformingtheDTproblemtotheDSTproblem,andthenusethealgorithmin[ 12 ]tosolveDSTproblem,thusobtainingthesolutionforDT.Notethatafterthetransformation,weneedtondtherightrootforapplyingthatalgorithm,sincetheratioforDSTismaintainedifandonlyifrisintheoptimalDT.Thiscanbedonebyenumeratetheneighborsofthenodewiththeminimumdegree,sinceatleastoneoftheneighborsshouldbeinoptimalDT.Therefore,fromLemma 11 ,wecanobtainthesameapproximationratioforDT.Finally,sincetheterminalnodesinG'arethosedummyverticeswhichhavenooutgoingedges,wecansimplyremovethesedummyverticesandthoseedgesincidenttothemtoobtainaDT.Bysettingi=lgn,thealgorithmwillobtainanO(lg2n)approximationinquasipolynomialtimewhichisnO(lgn). 91 PAGE 92 9 .Fromthehighlevel,thealgorithmconsistsofthefollowingmainsteps: 1. Initializeatree,whereeachvertexinagivengraphisaseparatesubtree. 2. CreateasortedlistofalledgesinGbyweights. 3. WhileallthesubtreesarenotmergedintoatreeDT Removeanedgewithminimumweightfromthesortedasaninactiveedge. (b) Whentheedgefromthelistconnectstwodifferentsubtreeswithoutanycircles. i. MergethemintoasubtreebyaddingtheinactiveedgetoDT. ii. Ifnewinternalnodesconvertedfromleafnodesduringmergingsubtreeshavesomesinglenodesubtreeneighbors,thenlinktheseinternalnodeswiththembyactiveedgessoastomaximizethesizeofleafnodesinDT. iii. Ifthenewinactiveedgeconnectingtoanodewhichalreadyhasanactiveedgeincidenttoit,removethisactiveedge. 4. PrunealltheleafedgesoftheresultantDT.ThedetailsofmergingtwosubtreesareshowninAlgorithm 10 .TheonlypurposeofactiveedgesistomaximizethenumberofleafnodeswithoutconsideringtheirweightsduetothefactthatthetotalweightofDTdoesnotincludetheweightsfromleafedges. 92 PAGE 93 53 whichdepictstheexecutionofheuristicalgorithm.IntheFigure 53 (b)and(c),sinceleafnodegisanewinternalnodewithsinglenodesubtreeiaftermerging,glinksitogetherbyactiveedge.However,activeedge(i,g)hastoberemovedwhennewinactiveconnection(c,i)occursconnectingsubtreectoi.Becauseallinternalnodeshavetobeconnectedbyinactiveedges,andifwedonotremoveactiveedge(i,g),iwillbeinternalnodewithanexistingactiveedge(i,g)andthusincreasingtheweightofDT. Figure53. TheexecutionofHeurDTalgorithm 93 PAGE 94 endwhile 1. 2. ThecomplexityofMergeSortalgorithmtosortedgesbyweightinStep2isO(mlogm)[ 17 ]. 3. InStep3,foreachnewinternalnodeufrommergingsubtrees,allitsneighborsneedtobecheckedtoseeifthereareanysubtreeswithonlyonenode,whichisO()foreachedgetoprocess.However,onlymergingsubtreeswouldbeabletoconvertleafnodesintointernalnodes,andthetotalnumberofmergingprocesstakesplaceatmostO(n).Asaresult,theoverallcomplexityforStep3isO(n). 4. Pruningallleafnodesonlytakeslineartime.Fromtheaboveanalysis,thetotalrunningtimecomplexityisdominatedbyStep2.Thatis,theruntimeofthisheuristicalgorithmisatmostO(n2logn),wherethegraphisdenseenough(O(n)andmn2). 94 PAGE 95 ifvhasonlyoneactivelinkinDTthen foreachneighbornodewofN(u)inGdo ifjTsub(w)j==1then endfor endif ifv==internalandvhasatleastoneinactivelinkthen foreachneighbornodewofN(v)inGdo ifjTsub(w)j==1then endfor endif 95 PAGE 96 BDTSize CRunningTime SimulationresultsforHeurDT,MSTLandoptimalresults Firstofall,letuspresenttheIntegerProgramming(IP)formulationoftheDTproblem.WewilluseCPLEXtosolvethisIPandtheoptimalresultwillbecomparedwiththatofourheuristic. 96 PAGE 97 54A illustratestheperformanceofthosethreeapproachesintermsoftheweightofDT.AsshowninFigure 54A ,theDTweightsfromHeurDTareveryclosetotheoptimalsolution,whichshowsHeurDTperformsextremelywell.Inparticular,theDTweightfromHeurDThasatmost8%ofadditionalweightsthantheoptimalresultswhenn=15accordingtotheFigure 54A .Inaddition,thegapinDTweightsbetweentheoptimalsolutionandthatofHeurDTdoesnotshow 97 PAGE 98 54B ,thedifferenceinsizebetweenoptimalsizeandtheDTobtainedbyHeurDTisverysmall.AccordingtotheFigure 54B ,theDTbuiltfromHeurDThasatmost1.2nodesmorethantheDTfromOptDTinthecaseofthenetworkinstancewith15nodes.Signicantly, 54B indicatesthatthedifferenceinDTsizeisnotaffectedbynetworksize.However,thesizeofMSTLismuchlargerthanthatofHeurDT.ItisverycleartoprovethatHeurDTmaximizingthenumberofleavesduringthemergingcandrasticallyreducetheDTsize.Asnincreases,thedifferenceinsizeofDTbetweenMSTLandtheHeurDTsignicantlyincreasesup,suchas33%additionalnodeswhenn=4and42%morenodesinMSTLwhenn=17thantheDTgeneratedbyHeurDT.ThisrevealssubstantialdifferenceofDTsizebetweenthemwhenwetakethetotalnumberofnodesinGintoconsideration.ComparingtheRunningTime.Figure 54C presentstherunningtimeofallthreeapproaches.Asexpected,therunningtimeforndingtheoptimalsolutionisextremelyhigh.Onemoretime,itconrmsthatitistooexpensivetondtheoptimalsolution,leadingtothestudyofapproximationsolution.Interestingly,therunningtimeofHeurDTandMSTLisveryclosetoeachother.Forexample,HeurDTtakesaround37msmoreinaveragethantheMSTLapproach.Thisisduetothetimespentonidentifyingactiveedgesandremovingthemiftheyareplacedbetweeninternal PAGE 99 99 PAGE 100 3 5 18 20 24 28 40 41 45 54 59 65 67 68 74 75 79 80 ]seekaminimumsizeCDS(MCDS),whichisNPhard[ 27 ],astheirmajordesigngoal.MinimizingthecardinalityofCDScanhelptodecreasethecontroloverheadsincebroadcastingforroutediscovery[ 33 51 ]andtopologyupdate[ 1 ]isrestrictedtoasmallsubsetofnodes[ 13 ].Thereforebroadcaststormproblem[ 73 ]inherenttoglobaloodingcanbegreatlydecreased.However,thereareseveralimportantfactorsthatneedtobefullyinvestigated.Therstimportantfactoristhenetworklatency,alsorepresentedasdiameterofCDS,whichisthelongestshortestpathbetweenanypairofnodesinCDS.Consideringthesituationthatthereceiverisnotwithinthetransmissionrangeofthesender,communicate 100 PAGE 101 68 75 81 82 ]haveaddressedthisissue.However,theyonlyconsideredthesizeofCDStogetherwiththefaulttolerance,withoutthediameterofCDSandrunningtime.ThemaincontributionofthisworkisthemultifactorsmodelforafaulttolerantMCDSwithboundednetworklatency(diameter)andthelowrunningtimeforafeasiblesolutionisexpectedaswell.Thecharacterizationofthismodelisthat(1)avariableisinvolvedinthisalgorithmasaninputtomaketheperformancetunable.(2)tradeoffsamongmultiplefactorsareshownthroughanalysis,whichhasnotbeenaddressesinrelatedworks.Morespecically,theproposedprogressivealgorithm,whichistheinputofourmodel,allowsforsystematicimprovement.Takinginspirationfromcomputationalgeometryandprobability,wedeviseimprovementtechniquesforsystematicallyreducing 101 PAGE 102 34 ],CDSBD[ 38 ]andBDA[ 88 ]underthesameparameters.TheresultsdemonstrateouralgorithmoutperformsCDSBDD,CDSBDandBDAinmosttestingcases. 15 ],whereallnodeshavethesametransmissionranges,andDiskGraphswithBidirectionalLinks(DGB)(wewillintroduceDGBinSection 6.3 ). 28 ],twopolynomialtimealgorithmstoconstructaCDSisproposedbytheauthors.Therstalgorithmhasperformanceratioof2(H()+1),whereHisaharmonicfunctionandisthemaximumdegreeofG.Theideaoftherstalgorithmistoidentifythenodewithamaximumdegreeastheroot.ThenbuildaspanningtreeTattheroot,growTuntilallnodesareaddedtoT.Then,allleafnodesarecutoffandtheremainingnodesinTareaCDS. 102 PAGE 103 19 20 59 ].In[ 54 ],Ruanetal.introducedanothercentralizedandgreedyalgorithmofwhichtheperformanceratiois(2+log).WuandLi[ 79 ]proposedanalgorithmthatcanquicklygenerateaCDSbasedontheconnectivityinformationwithinthe2hopneighbors.Thisapproachusesamarkingprocess.Inparticular,eachnodeismarkedtrueifithastwounconnectedneighbors.AllthemarkednodesformaCDS.TheauthorsalsointroducedsomedominantpruningrulestoreducethesizeoftheCDS.In[ 74 ],theauthorsshowedthattheperformanceratioof[ 79 ]iswithinafactorofO(n)wherenisthenumberofnodesinanetwork. 3 5 74 ],theauthorspresentedadistributedalgorithmwithaconstantperformanceratioof8.Later,Cardeietal.presentedanotherdistributedalgorithmin[ 11 ].Thisalgorithmhasthesameperformanceratioaspreviouswork.However,themessagecomplexityislowerthanthatof[ 74 ].Asweknowthatdistributedalgorithmhasabetterperformancethanlocalizedalgorithms.Inthelocalizedalgorithms,in[ 4 ],Alzoubietal.proposedalocalizedalgorithmswithaperformanceratioof192.Althoughtheperformanceof[ 4 ]cannotcompetewiththatof[ 74 ]and[ 11 ].Theiralgorithmonlyneedonehopneighborsinformation.Therefore,onceanodeknowsthatithasthesmallestIDamongitneighbors,itbecomesadominator.Then,thedominatorscanbeconnectedbytheintermediatenodesinthenextstep.In[ 41 ],Lietal.proposedanotherlocalizedalgorithmwithaperformanceratioof172,whichisbetterthan[ 4 ]. 103 PAGE 104 11 74 ]areapplicableinDGB.In[ 67 ],Thaietal.rstproposedtheperformanceratioofCDSonsizeinDGBandthetwoproposedalgorithmscanbeimplementedbydistributedways.However,theonlydifferencebetweentwoalgorithmsisthestrategytoselectMIS,therstalgorithmemployedWan'salgorithm[ 74 ]tochoosethenodesinMIS,whilethesecondalgorithmusedthegreedystrategy,thatistoincludetheminimumnumberofnodesinMIS,thusleadingtoabetterperformancethantherstalgorithm. 45 ].However,theydidnotgiveaguaranteedperformanceintheirmodel.In[ 38 ],Lietal.studiedtheCDSproblemwithboundeddiameterinUDGandproposedaconstantapproximationalgorithm,calledCDSBD.However,theiralgorithmiscentralizedandnoexperimentalresultsareprovided.Asanextendedworkof[ 38 ],Kimetal.rstmadetheircentralizedalgorithmtobedistributed,thenaddedenergyconsiderationwhenconstructedtheCDS.Simulationresultsandcomparisonagainstotherrecentalgorithmswerereportedattheend.Theproblemin[ 34 ]isthattheyemphasizedthattheUDGcannotbeusedasnetworkmodels,sincethetransmissionrangesofallnodesmaybedifferent.However,theystillusedUDGastheirmodelthroughtheirwholework.Incontrast,wewillemployanewnetworkmodel,DGB,tostudyourmodelinthiswork.Insummary,noneofthepreviousworkhaveaddressedthefollowingissues.First,theperformanceofourmodelistunable,itcanbeadjustedinarangebyanuserdenedinput.Second,wendoutthetradeoffsamongCDSsize,diameter,runningtimethroughtheoreticalanalysisandsimulation,thatis,itishardtooptimizethesefactorsatthesametime.Third,runningtimeisrstlyintroducedasametric 104 PAGE 105 ln(2cos( 67 ],wheretheindependentneighborsofanodeuaredenedasasetofnodesthatadjacenttousatisfyingthatanytwonodesinthesetareindependent. 105 PAGE 106 61 showsanexamplefortheintroducedterminologies. Figure61. Givennoderastheroot,nodea,b,f,gareterminals.ristheparentofcande.Nodea,baresiblings,f,garesiblings.Nodea,b,f,garer's2hopsawayneighbors 106 PAGE 107 68 81 82 ]hasaddressedthegeneralfaulttolerantMCDSproblem.However,noneofthemmentionedhowtobounditsdiameter.Inourpreviouswork[ 68 ],wehaveproposedasolutionforkmCDSproblem,where1km+1,asillustratedinAlgorithm 11 .Inthischapter,westillusethisalgorithmtosolveourmultifactormodel.However,anewanalysisisproposedforthediameterofkmCDS.ThemainideaofAlgorithm 11 isthatmergingallthek'blocksin1ConnectedmDominatingSet(1mCDS)intoonlyonek'blockbyaddingextranodes,wherek'=2initially.Then,weincreasek'by1andrepeattheaboveoperationuntilk'=k.Wecanuseany1CDSwithboundedsizeanddiameterastheinputofAlgorithm 11 .However,inordertomakethesolutionadjustablebytheuser,an(,)CDS,tobeintroducedinSection 6.6 ,ispreferredtobeaninputofAlgorithm 11 68 ] 68 ] 107 PAGE 108 11 producesakmCDSwith(2K+2m+1)approximationonsize,whereC1mandCkmarethe1mCDSandkmCDSwithoptimalsolutiononsizerespectively.Proof:CkmistheunionofC1mandthenodesaddedintoC1m,inordertomakeC1mkconnected.ThenumberofnodesweaddedtomakeC1mkconnectedisatmost2(k2)(jC1mj1)+2(K+1)(jC1mj1)[ 68 ].Therefore, 68 ],wealreadyconcludedthefollowinginequality,jC1mjjCDSj+(K+m1)jC1mjThus, 108 PAGE 109 12 and 13 ,wehavethefollowinginequality:d(Ckm)d(C1m)+2d(C11)+2+2D+4Dkm+42 11 ,wewouldliketointroduceanalgorithmtodetermineaCDSwithabetterapproximationondiameterthanexistingwork.Asweknow,theauthorsin[ 38 ]presenteda3approximationalgorithmondiameter,whichwasthebestknownresultatthattime.Inthissection,analgorithmthatguaranteesCDSwith2approximationondiameterofCDSisdescribedandthesizeofCDSisboundedaswell.Thedifferencebetweenotherexistingwork[ 34 68 ]andouralgorithmisthattheytrytominimizetheCDSsizewhilethediameterisbounded.Incontrast,wewanttominimizetheCDSdiameterwhilethesizeisbounded.ToconstructaCDS,weoftenemployanMaximalIndependentSet(MIS)whichisalsoasubsetofallthenodesinthenetwork.ThenodesinMISarepairwisenonadjacentandnomorenodescanbeaddedtopreservethisproperty.Therefore,eachnodewhichnotinMISisadjacenttoatleastonenodeinMIS.Thus,anMISisindeedaDS.IfthenodesinMISareconnectedbyaddingmorenodestotheMIS,aCDScanbeconstructed.Here,ouralgorithmconsistsofthefollowingthreephases, 1. Rootrisrandomlychosenandweonlyselectthenodes,e.g.nodey,intoMIS,wheredryisaneven.Then,thenodesinMISiscoloredblueorblack,andallothernodesarecoloredgrayorred. 109 PAGE 110 SwapthecolorsofblueandrednodesandchangesomenodestoblackaccordingtotherulesdescribedinsecondphaseofAlgorithm 12 .Consequently,thebluenodesandblacknodesmaybeadjacent,buttheystilldominateothernodes,thereforetheMISwithblackandbluenodesischangedtoaDS. 3. ConnecttheDSwithsomeintermediatenodes,andDSwillbeaCDSnally.ThedetailsofouralgorithmareshowninAlgorithm 12 .Formoreclarication,weshowanexampletoillustratethesecondphase. PAGE 111 62 presentsanexamplethatillustratestheprocedureofswappingcolorinsecondphase.Inthisexample,weassumethatlevela+1isequaltok,whereaiseven,andtheMISwithblueandblacknodesisproducedinrstphase.Now,wedescribehowtoswapthecolorsstepbystep. 1. TheinitialsituationisshowninFigure 62 (a),whereblackandbluenodesareinMIS. 2. Inline3ofsecondphase,step(1)isshowninFigure 62 (b),thetworednodesinlevela1arechangedtoblue,andnodexischangedtored. 3. Forstep(2),inGa,yistherednodedominatedbyx,coloroney'sneighborpinGa1black.ThissituationisshownasinFigure 62 (c). 4. Forstep(3),therednodezinGa+1isdominatedbyx,colorzblack.ThisisshowninFigure 62 (d). 5. Inline6,allgraynodesarecoloredblackandallrednodesarecoloredgray,seeFigure 62 (e).Atthismoment,thereareonlyblackandgraynodesinthegraphandforeachblacknode,wecanalwaysndexactlyonenodethatmakeitconnectedwithotherblacknodesinupperlevelwithin2hops,sothenumberofblacknodeforconnectingtheDSisatmostjDSj1. 12 ,thenjTCDSj2(K1)KjCDSj1andd(TCDS)2D+6inaDGB.Proof:ItisknownthatforanMISIinDGB,jIjKjCDSj[ 67 ].However,whenweswapthecolorsofnodes,theMISischangedtoaDS,so,jDSj(K1)jIjsinceintheworstcase,inGi,foreveni,ifwechangethecolorforeachbluenodex,wehavetochangeatmostKnonblacknodestoblacktomaintainthewholenetworkdominatedbyallblacknodes,suchthatxisinMISandeachnodeinMIScanbeadjacentatmostKindependentneighbors[ 67 ].Therefore,thedifferenceinsizebetweenMISandDSisjDSj(K1)jIj.Thus,jTCDSjjDSj+thesizeofconnectingnodes2jDSj12(K1)jIj12(K1)KjCDSj1.FordiameterofCDS,suppose 111 PAGE 112 B C D E Anexampleforsecondphaseinalgorithm 12 112 PAGE 113 34 38 ]canguaranteetheCDSsizeanddiameter,theirperformanceratiosarexed.Consideringtheexibilityofwirelessadhocnetwork,weintroduce(,)CDSintoourmodelsothatrst,itsperformanceratiosaretunablebasedontheinput.Second,thecenterofnetworkisinvolvedintheCDSconstructiontoenhancetheperformance.Third,itcanconstructaCDSwithapproximatelysatisfyingthesizeconstraintanddiameterconstraint.Asweintenttobalancethesizeanddiameter,thedenitionof(,)CDSingiveninwirelessadhocnetworksasfollows: 1. (Size)ThesizeofCisatmosttimestheminimumCDSsize. 2. (Diameter)ForanypairofvertexuandvinC,d(C)isatmosttimestheminimumdiameterofCDSplusaconstantnumber.In(,)CDS,isanuserdenedinput,andusuallyisafunctionof.Therefore,thevalueofdependsontheuserdenedinput.Inthefollowing,wewilldescribehowtogeneratean(,)CDS,analyzethetimecomplexityandpresentthetradeoffbetweenthesizeanddiameterthroughanalysis.ThegeneralideaofPAisasfollows. 1. Rootrshouldlocateatthecenterofnetwork,whichisthemidpointofthelongestshortestpathbetweentwonodesinG. 2. ConstructaCDSTCDSrootedatrbyusingBDA[ 88 ]inourpreviouswork,whereBDAisanapproximationalgorithmforCDSwith2Kapproximationonsizeand4approximationondiameter. 3. ConstructaShortestPathTree(SPT)TSPTrootedatr,whichonlyincludesalltheshortestpathsfromrtoeveryothernodeinTCDS. 4. TraverseTCDSinadepthrstmanner.Whenvisitinganodeu,ifthenumberofhopsfromrtouinTCDSislargerthanauserdenedthresholdtimesthe 113 PAGE 114 46 ]toconstructanSPTTSPT PAGE 115 RootSelectionandCDSTreeConstruction:WithDistributedSPTalgorithm[ 46 ],eachnodemaintainsaglobalvariable,whichstoresthecurrentlongestshortestpathinthegraphG,ifalongershortestpathisfound,theglobalvariableofeachnodewillbeupdated.Attheend,wecouldndthemidpointofgloballongestshortestpath.WhileconstructingTCDSrootedatrbyBDA[ 88 ],eachnodeuneedstomaintainapointer[u]foritsparentonthetreeTCDSandanupperboundd[u]forthenumberofhopstor.WeusetheINITIALIZEandRELAXalgorithmsin[ 39 ]toinitializeandmaintainbothoftheseattributes. 2. ShortestPathTreeConstruction:TSPTrootedatrisconstructedbyusingDistributedSPTalgorithm.ItonlycontainsalltheshortestpathsfromtherootrtoeveryothernodeinTCDS. 3. DepthFirstSearch(DFS):TraversetheTCDSinaDFSmannerbeginningfromtherootralongthepathsfromrtoalltheothernodesinTCDS.Whennodeuisreachedforthersttime,ifd[u]isgreaterthanDSPT(r,u),thentheshortestPruinTSPTisaddedtoTCDSandd[u]and[u]areupdated.Afterthis,nodeu'sparentvneedstobecheckediftheupdatedpathfromrtouwillresultinreducingthenumberofhopsfromrtov.Ifso,thenv'sparentwillbecheckedandsoonuntiltherootrisreached.WiththeexecutionofBDA,distributedSPT(dSPT)(e.g.[ 46 ]),anddistributedDFS(dPFS)(e.g.[ 57 ]),TCDS,TSPTandaDFStraversalordercouldbeachieved.ThedetailsofPAareillustratedinAlgorithm 13 .ToevaluatethecorrectnessofthePA,weexaminewhetherthetwoconstraintsinthedenitionhasbeensatised.Takingasanuserdenedinput,wederivearelationshipbetweenand,whichshowstherelationshipbetweenthesizeoftheconstructedCDSandtheoptimalsolutionofCDSonsize.WealsoanalyzethetimecomplexityofthePA.Denew(TCDS)asthetotalweightofTCDSinG,whereweassumeeachedgehasbeenassignedtheunitweightof1.ThenDSPT(u,v)andDCDS(u,v)areequaltotheweightofTSPT(u,v)andTCDS(u,v)respectively.AnotherobservationisthatjTCDSj=w(TCDS)+1,sincethenumberofnodeinatreeequalstothetotalnumberofedgesplus1,whichalsoequalstow(TCDS)+1.Meanwhile,aswementionedbefore,thelowerboundofminimumdiameterofCDSisD2.Actually,theupperboundfortheminimumdiameterofCDSisD,i.e.,allthenodesinGareinCDS,therefore,D=D. 115 PAGE 116 63 .Therefore,hD+1,thenthemaximumnumberofhopsbetweenuandvisatmost2h,thatis2(D+1).2Inrealwirelessadhocnetwork,case(2)rarelyhappens,sinceitrequiresthatallnodesareuniformlydeployedasaring.However,inmostcases,theyaredeployedrandomly.Therefore,thediameterofCDSreturnedbyPAisboundedby(D+2)inmostcases. 116 PAGE 117 AllthenodesintheringareaCDSwithdiameterof8 PAGE 118 67 ],wehave:Psize(5)K PAGE 119 88 ]andLemma 16 88 ]anddSPTanddDFSrunatmostO(n2)timecomplexityandsendO(n2)messages[ 46 ][ 57 ].Now,weanalyzetheprocedureofndingthecenterofnetwork.ThedSPTalgorithmisexecutedateachnodexsimultaneously,afterthat,xneedstobroadcastthelongestpathinSPTrootedatxandcompareitwiththelongestpathsreturnedbyothernodes.Therefore,thisprocedureneedsO(n2)timecomplexityandO(n2)messagecomplexity.SinceallotheroperationonlytakeatmostO(n)timecomplexityandO(n)messagecomplexity,theoverallmessagecomplexityandtimecomplexityofPAareO(n2)andO(n2).2 6.6 allowsaCDStobeconstructedwithguaranteedandtunableperformance,whilethecenterofnetworkgreatlyhelpstoimprovetheperformance,e.g.reducingthesizeanddiameterofCDS,whichwillbeveriedinsimulation.However,the 119 PAGE 120 44 ],wepresentthreedistinctimprovements:reducingmultiplepaths,removingredundantterminalsandlocatingcentralarea.Therstandsecondtechniquesreducethesize,whereasthelasttechniquereducestherunningtime. 64 .InFigure 64 (a),theblackandbluenodesareinTCDSandblacknodesrepresenttheMIS.Supposeweset=1,therednodesthatconsistoftheshortestpathfromrtovisaddedintoTCDS.TheredundantnodeinTCDSisu,sincevcanbeconnectedtotherootviashortestpath.InFigure 64 (b),theredundantnodeuisremovedfromTCDS,andthenewCDSissmallerthantheoriginalone.Bydoingthis,weareabletoreducetheCDSsize,leadingtothelowmessageoverheadandtransmissionerrorwithoutinterferingwithCDSdiameter.Wewillshowtheeffectivenessofthistechniqueinsimulation. 120 PAGE 121 B Anexampleforreducingmultiplepath 14 isshowntoimplementthistechnique.Foreasyunderstanding,Figure 65 showthesituationsbeforeandafterapplyingthetechnique.InFigure 65 (a),theblacknodesareinTCDS,whiletheredundantterminalinTCDSisu,since,besidesu,xisalsodominatedbyv.InFigure 65 (b),the 121 PAGE 122 redundantterminaluisremovedfromTCDS,andthenewCDSissmallerthantheoriginalone. B Anexampleforremovingredundantterminals 122 PAGE 123 44 ],eachlinkbetweentwonodesisassignedaweight.Consideringalink(u,v)innetwork,themeasureofweightonlink(u,v)isgivenbythenumberofnodeswithinthetransmissionrangeofnodesuandv(otherthanuandv).LetD(u,v)denotethediskcenteredatuandradiusju,vj.Thentheweightw(u,v)ofthelinkisdenedintermofthenumberofnodesindisksD(u,v)andD(v,u)asdenedbelow. 50 ]toapproximatethecentralareaofthegivennetwork.Weemployatreeanditsleafnodesaredenedastheconvexhullofthegivennetwork.Supposetherootofthetreeisintheconvexhull,moreinternallinkandveryfewboundarylinksareincludedinthetree.ThisapproachhasthedrawbackofdecreasingtheprobabilityoftherootofCDSlocatedatthecentralarea.Toovercome 123 PAGE 124 6 .Observation:Theweightonboundarylinkisverylikelylessthantheweightoninternallink.Fromtheaboveobservation,itisnotdifculttoseethatthecentralareaismoredense(thenumberofnodeperarea)thantheboundaryofnetwork.Therefore,theweightonboundarylinkisnormallylessthanthatoninternallink.In[ 44 ],anO(n2lgn)algorithmforconstructingMinimumInterferenceTree(MIT),whichisbasedonthestructureofMST,isreported.ThisalgorithmisamodicationofthewellknownKruskal'salgorithmforconstructingMST[ 17 ]thatincreasestheprobabilityofboundarylinksincludedinMIT.Wecanusethisalgorithmtoconstructourtreeandtheleafnodesaredenedasthenodesetofconvexhull.Finally,wepickuponenodeinconvexhullastherootofCDS.WecanalsoobservefromthetreeinFigure 66 that,asexpected,mostofthelinksneartheboundaryofnetworkarepresentedinourtree,causingmostofleafnodeslocatedatthecentralarea. Figure66. Anexampleofleafnodeslocatedatcentralarea.Theblacknodesconsistofthetree. Sincethereisstillsmallprobabilitythattheleafnodeisattheboundary,notincentralarea,ifweselectsuchnodeastherootofCDS,thenourtechniquewillfailattheend.Inordertominimizetheprobabilityofthisunexpectedsituation,wemaysortthetheleafnodesbasedonthedegreeofeachnodeandthenpickupthenodewith 124 PAGE 125 67 givesagoodexample.InFigure 67 ,theweightofeachlinkisassignedbyDenition 6 ,andthedottedlinksdenotetheconstructedtree,whereaandbaretheleafnodes.IfweselectbastherootofCDS,thenthediameterofCDSwillbe5.Ontheotherhand,ifaistheroot,thediameterwillbeonly4,sinceaisincentralarea,butbisnot.Basedonouraboveidea,wewillonlypickupaastherootsincethedegreeofaislargerthanthatofb. Figure67. AnexampleofselectingthenodewithmaximumdegreeastherootofCDS 34 ],CDSBD[ 38 ],BDA[ 88 ]andPA.Asfarasweknow,CDSBDD,CDSBDandBDAaretheexistingalgorithmsthatguaranteetheconstantperformanceratiosonsizeanddiameter,andwewillshowthatPAoutperformsCDSBDD,CDSBDandBDAinmosttestingcases.Second,wealsowouldliketoverifytheimportanceoflocatingthecenterofnetworkandtesttherunningtimeofBDAandPAtoshowthetradeoffbetweenperformanceandrunningtime.Third,wedoexperimentsby 125 PAGE 126 11 bycomparingwithPAsothatthetradeoffbetweenthefaulttoleranceandsizecouldbesystematicallydiscovered.In[ 34 ],theauthorrstproposedtheAverageBackbonePathLength(ABPL)asanotherfactortoevaluatetheCDS.ABPLofaCDShasbeendenedasthesumofhopdistancebetweenanypairoftwonodesinCDSdividedbythenumberofallpairofnodes.Inoursimulation,weevaluateABPLinadditiontodiameterandsize,sincethediameteronlyrepresentstheworstcasepathlengthofCDS,ignoringtheaveragepathlength,whileABPLcapturesaveragepathlengthformessagedelivery.Therefore,itisourintereststomeasuretheABPLofCDS.Tosimulatethenetwork,werandomlydeployednnodestoaxedareaof1,000mx1,000m.nchangedfrom10to100withanincrementof5.Eachnodevirandomlychosethetransmissionrangeri2[rmin,rmax]wherermin=100mandrmax=200m.Foreachvalueofn,1,000networkinstanceswereinvestigatedandtheresultswereaveraged. 38 ].ItselectsarootrandomlyandspansaCDSfromtheroot.TheapproximationratiosofCDSBDare11.4and3onsizeanddiameterrespectively.Ontheotherhand,theauthorsin[ 34 ]proposedCDSDBDthatcanbeimplementedindistributedway,however,itonlyguarantees4approximationondiameter,but6.906approximationonsize.Forthepurposeoffairness,weset=3(theapproximationratioofPAondiameter)inPA.Figure 68A showsthatthediametersofCDSbuiltbythethreealgorithmsareeasytodistinguish,sincethegapiscleartoobserve.Also,wenoticethatPAoutperforms 126 PAGE 127 BComparethesizeofCDS CComparetheABPLofCDS PerformanceforCDSBD,CDSBDDandPA CDSBDDbecauseofthelowerapproximationondiameterinPA,andCDSBDalsoperformsbetterthanCDSBDD.Figure 68B providestheperformancecomparisonofthethreealgorithmsonthesizeofCDS.ItshowsPAalwaysconstructsaCDSwithsmallersizethanCDSBDandCDSBDD,whichismuchbetterthantheoreticalanalysiswegaveinSection 6.6 .Asexpected,itisreasonablethatCDSBDDperformsbetterthanCDSBD,sinceCDSBDaddsmorenodesinCDStoshortenthediameter,whichwillcausetheincreaseonsize,butthegapbetweenthetwocurvesisnotlarge.Therefore,theperformanceofPAissatisfactoryonCDSsize.InFigure 68C ,asthenumberofnodesinnetworkincreases,theCDSreturnedbyPAalwayshasalowerABPLthanothertwoCDSsdeterminedbyCDSBDDand 127 PAGE 128 BComparethesizeofCDS CComparetheABPLofCDS PerformanceforBDAandPA Thepurposeofthissimulationistoverifytheimportanceofrootselectionandthetradeoffbetweenperformanceandrunningtimeatthesametime.Inordertohighlighttherootselection,weuseavariationofBDA,calledBDAMid,asareference.ComparedtoBDA,BDAMidselectsthecenterofnetworkastherootinsteadofchoosingrandomly.Also,weincludePAinthissimulationandissetto1.Figure 69A comparesthediameterofCDSconstructedbythethreealgorithms.Itisshownthat,underdifferentnumberofnodesdeployedinnetworks,theCDSbuiltbyPAhasthesmallestdiameter.WeobservethatthegapbetweenBDAandBDAMid 128 PAGE 129 69B ,wepresentthesizeofCDSobtainedfromallthreealgorithms,dependingonthenumberofnodesdeployed.ThesizesofCDSsreturnedbythethreealgorithmsareclosetoeachotherandtheyallincreasewiththenumberofnodes.Also,consideringthesamenumberofnodes,BDAreturnsalargersizeofCDSthanPAandBDAMid,althoughthegapsbetweenthesealgorithmlooksmallinFigure 69B .ThisillustratesthatthesizeofCDScanbereducedbychoosingthecenterofnetworkastheroot.AsshowninFigure 69C ,PAachievesaCDSwithsmallestABPL,whereasBDAMidstillperformsbetterthanBDA.Overall,PAleadstheperformanceonsize,diameterandABPLduetothecenterofnetwork.Therefore,itappearstobeanimportantissueintheconstructionofCDS.Table 61 summarizestherunningtimeunderdifferentnumberofnodes.Asthecomplexityanalysisindicates,theruntimeofBDAMidandPAismuchlongerthanthatofBDA.Thisisduetothelongtimespentondetectingthecenterofnetwork.Moreover,weshowinTable 61 thattheBDAMidstillrunsfasterthanPA,sincePAneedstocomputeTSPTtoshortenthediameter.Whenthenumberofnodesincreases,PAandBDAMidspendmoretimeondetectingthecenterofnetwork.Therefore,itisatradeoffbetweenthesize/diameterofCDSandrunningtime. 129 PAGE 130 BDA BDAMid PA PA3rdTech. ofNode Runtime Runtime Runtime Runtime 10 0.0001 0.0003 0.0006 0.0003 15 0.0002 0.0012 0.0014 0.0003 20 0.0003 0.0044 0.0046 0.0004 25 0.0004 0.0108 0.0132 0.0032 30 0.0006 0.0236 0.0280 0.0056 35 0.0007 0.0442 0.0536 0.0108 40 0.0012 0.0768 0.0980 0.0232 45 0.0012 0.1222 0.1558 0.0352 50 0.0013 0.1836 0.2372 0.0556 55 0.0014 0.2812 0.3742 0.0956 60 0.0030 0.4016 0.5418 0.1438 65 0.0036 0.5414 0.7378 0.2010 70 0.0040 0.7552 1.0332 0.2832 75 0.0046 0.9842 1.3488 0.3712 80 0.0046 1.3050 1.8086 0.5110 85 0.0050 1.6676 2.3030 0.6450 90 0.0045 2.1294 2.9578 0.8400 95 0.0048 2.6672 3.6630 1.0078 100 0.0060 3.7564 5.2224 1.4856 Table61. Runtime(ms) BComparethesizeofCDS Performancebasedondifferent 610 130 PAGE 131 610A ,eachlinerepresentsthediameterofCDSbasedononeofdifferentvaluesof.Whenissetto1,PAaddsashortestpathfromvtorifDCDS(r,v)islargerthanDSPT(r,v).Therefore,PAwith=1returnsaCDSwiththesmallestdiameter.Whenissetto4,PAonlyaddsthepathfromvtorinTCDSundertheconditionthatDCDS(r,v)isgreaterthan4timesofDSPT(r,v).Thus,theCDSbyPAwith=4hasthelargestdiameter.For=2,thecorrespondinglineisinthemiddle.Therefore,asweexpected,thediameterofCDSbuiltbyPAcouldbecontrolledbyadjustingthevaluesof.InFigure 610B ,eachlinerepresentsthesizeofCDSbasedononeofdifferentvaluesof.Whenissetto1,ifDCDS(r,v)islargerthanDSPT(r,v),PAaddsashortestpathfromvtor.Thisstrategywillincurmorenodestobeadded.Ontheopposite,whenissetto4,PAresultsinaCDSwithsmallersize.For=2,thecorrespondinglineisinthemiddle,thesamesituationasinFigure 610A .Inconclusion,theperformanceofPAcanbebalanceddependingonthevalueofandthetradeoffbetweensizeanddiameterisclear. BComparethesizeofCDS Performancefortherstandsecondimprovementtechniques Inthissection,weareinterestedinevaluatingtheeffectivenessofthepresentedimprovementtechniques.Sincetherstandsecondtechniquesaredevotedinto 131 PAGE 132 611A describestheperformanceintermsofthediameterofCDS.Asweexpected,thediameterisnotaffectedbytherstandsecondtechniques.Meanwhile,asobservedfromFigure 611B ,thesizeofCDSisreducedwhenthetwotechniquesapply.Clearly,webelievethattherstandsecondimprovementtechniquesareeffectiveinreducingthesizeofCDS. BComparethesizeofCDS Performanceforthethirdimprovementtechniques Table 61 alsosummarizesthecomparisonofruntimebyPAwithandwithoutthethirdimprovementtechnique.Incredibly,thethirdtechniqueachievesthereductionofrunningtimegreatly,althoughhereitsacricesalittleperformanceonsizeanddiameter,asshowninFigure 612A and 612B ,whichindicatesthetradeoffbetweenrunningtimeandsize/diameter.However,itisstillpromisingtondthecentralareainnetwork,inordertoachievethefastconstructionofCDS. 11 .WeintendtoillustratethatAlgorithm 11 improvesthefaulttoleranceofCDSbyaddingmarginaloverhead(intermsofthenumberofnodesaddedintoCDS).WetaketheCDSgeneratedbyPAastheinputofAlgorithm 11 ,andwesetk=2,m=1and=2. 132 PAGE 133 BComparetheSizeofCDS Performanceformultifactorsmodel Figure 613A comparestheperformanceofAlgorithm 11 andPAintermsofthediameterofCDS.Asweexpected,thereislittledifferenceonthediameterofCDSbasedonthetwoalgorithms,whichperfectlymatchesourtheoreticalanalysisforthediameterofkmCDS.Therefore,Algorithm 11 enhancesthefaulttoleranceofCDSwithoutaffectingitsdiametergreatly.Meanwhile,asobservedfromFigure 613B ,thesizeofkmCDSobtainedfromAlgorithm 11 iscertainlylargerthanCDSbyPA.Specically,theperformanceofthetwoalgorithmsisrelativelyproportional.Asobservedfromourexperiments,thesizeofkmCDSobtainedfromAlgorithm 11 isalmost1.1timesthesizeofCDSreturnedbyPA.TheresultsindicatethatconsideringthefaulttolerancewillincreasethesizeoftheCDSatthesametime.However,theincreaseinsizeisstillboundedandpredictable.Therefore,itiscleartoseethetradeoffbetweenthefaulttoleranceandsize. 133 PAGE 134 134 PAGE 135 135 PAGE 136 136 PAGE 137 [1] [2] Acharya,MithunandThuente,David.IntelligentJammingAttacks,CounterattacksandCounterAttacksin802.11bWirelessNetworks.ProceedingsofOPNETWORK.2005. [3] Alzoubi,KhaledM.,Wan,PengJun,andFrieder,Ophir.DistributedHeuristicsforConnectedDominatingSetsinWirelessAdHocNetworks.JournalofCommunicationsandNetworks4(2002):22. [4] Alzoubi.,KhaledM.,Wan,PengJun,andFrieder,Ophir.MessageOptimalConnectedDominatingSetsinMobileAdhocNetworks.MobiHoc'02:Proceedingsofthe3rdACMinternationalsymposiumonMobileadhocnetworking&computing.NewYork,NY,USA:ACM,2002,157. [5] Alzoubi,KhaledM.,Wan.,PengJun,andFrieder,Ophir.NewDistributedAlgorithmforConnectedDominatingSetinWirelessAdHocNetworks.HICSS'02:Proceedingsofthe35thAnnualHawaiiInternationalConferenceonSystemSciences(HICSS'02)Volume9.Washington,DC,USA:IEEEComputerSociety,2002,297. [6] Arkin,EstherM.,Halldorsson,MagnusM.,andHassin,Refael.ApproximatingTheTreeAndTourCoversOfAGraph.1993. [7] Baruch,Awerbuch,Andrea,Richa,andChristian,Scheideler.AJammingResistantMACProtocolforSinglehopWirelessNetworks.PODC'08:ProceedingsofthetwentyseventhACMsymposiumonPrinciplesofdistributedcomputing.NewYork,NY,USA:ACM,2008,45. [8] Bellardo,JohnandSavage,Stefan..11DenailofServiceAttacks:RealVulnerabilitiesandPracticalSolutions.Proceedingsofthe12thconferenceonUSENIXSecuritySymposium.2003. [9] Bomze,ImmanuelM.,Budinich,Marco,Pardalos,PanosM.,andPelillo,Marcello.TheMaximumCliqueProblem.HandbookofCombinatorialOptimization.KluwerAcademicPublishers,1999,1. [10] Bron,CoenandKerbosch,Joep.FindingAllCliquesofanUndirectedGraph.Commun.ACM16(1973).9:575. [11] Cardei,Mihaela,Cheng,Xiaoyan,Cheng,Xiuzhen,andzhuDu,Ding.ConnectedDominationinMultihopAdHocWirelessNetworks.InProc.the6thInternaConferenceonComputerScienceandInformatics.2002. [12] Charikar,Moses,Chekuri,Chandra,yatCheung,To,Dai,Zuo,Goel,Ashish,Guha,Sudipto,andLi,Ming.ApproximationAlgorithmsforDirectedSteinerProblems.JournalofAlgorithms.1998,73. 137 PAGE 138 Cheng,Xiuzhen,Ding,Min,Du,Hongwei,andXiaohua,Jia.OntheConstructionofConnectedDominatingSetinAdhocWirelessNetwork.inSpecialIssueonAdHocWirelessCommunicationsandMobileComputing.2004. [14] Chiang,JerryT.andHu,YihChun.DynamicJammingMitigationforWirelessBroadcastNetworks.INFOCOM2008.The27thConferenceonComputerCommunications.IEEE.2008,1211. [15] Clark,BrentN.,Colbourn,CharlesJ.,andJohnson,DavidS.UnitDiskGraphs.DiscreteMath.86(1990).13:165. [16] Codenotti,Paolo,Sprintson,Alexander,andBruck,Jehoshua.AntiJammingSchedulesforWirelessDataBroadcastSystems.InformationTheory,2006IEEEInternationalSymposiumon(2006):1856. [17] Cormen,ThomasH.,Leiserson,CharlesE.,Rivest,RonaldL.,andStein,Clifford.IntroductiontoAlgorithms.TheMITPress,2001,2ndrevisededitioned. [18] Dai,FeiandWu,Jie.OnConstructingkConnectedkDominatingSetinWirelessNetworks.InProceedingsofthe19thInternationalParallelandDistributedProcessingSymposium(IPDPS).2005. [19] Das,BevanandBharghavan,Vaduvur.RoutinginAdHocNetworksUsingMinimumConnectedDominatingSets.InternationalConferenceonCommunications.1997,376. [20] Das,Bevan,Sivakumar,Raghupathy,andBharghavan,Vaduvur.RoutinginAdHocNetworksUsingaSpine.InternationalConferenceonComputersandCommunicationNetworks.1997,1. [21] Desmedt,Yvo,SafaviNaini,Rei,Wang,Huaxiong,Batten,Lynn,Charnes,Chris,andPieprzyk,Josef.BroadcastAntijammingSystems.Networks,1999.(ICON'99)Proceedings.IEEEInternationalConferenceon.1999,349. [22] Du,DingZhuandHwang,FrankK.CombinatorialGroupTestinganditsApplications(2nded.).WorldScientic,Singapore,1999. [23] Du.,DingZhuandHwang,FrankK.PoolingDesigns:GroupTestinginMolecularBiology.WorldScientic,Singapore,2006. [24] Du,DingZhu,Thai,MyT.,Li,Yingshu,Liu,Dan,andZhu,Shiwei.StronglyConnectedDominatingSetsinWirelessSensorNetworkswithUnidirectionalLinks.APWeb.2006,13. [25] Feige,Uriel.AThresholdofLnNforApproximatingSetCover.JournaloftheACM45(1998):314. 138 PAGE 139 Felstead,BarryE.FollowerJammerConsiderationsforFrequencyHoppedSpreadSpectrum.MilitaryCommunicationsConference,1998.MILCOM98.Proceedings.,IEEE.vol.2.1998,474. [27] Garey,MichaelR.andJohnson,DavidS.ComputersandIntractability.AguidetotheTheoryofNPcompleteness.NewYork,NY,USA:Freeman,1979. [28] Guha,SudiptoandKhuller,Samir.ApproximationAlgorithmsforConnectedDominatingSets.Algorithmica20(1996):374. [29] Gupta,RajarshiandWalrand,Jean.ApproximatingMaximalCliquesinAdhocNetworks.Personal,IndoorandMobileRadioCommunications,2004.PIMRC2004.15thIEEEInternationalSymposiumon1(2004):365. [30] Hang,Wang,Zanji,Wang,andJingbo,Guo.PerformanceofDSSSagainstRepeaterJamming.Electronics,CircuitsandSystems,2006.ICECS'06.13thIEEEInternationalConferenceon(2006):858. [31] Hassan,AmerA.,Hershey,JohnE.,andSchroeder,JamesE.OnaFollowerToneJammerCountermeasureTechnique.Communications,IEEETransactionson43(1995).234:754. [32] Hassan,AmerA.,Stark,WayneE.,andHershey,JohnE.FrequencyHoppedSpreadSpectruminthePresenceofaFollowerPartialbandJammer.Communications,IEEETransactionson41(1993).7:1125. [33] Johnson,DavidB.andMaltz,DavidA.DynamicSourceRoutinginAdHocWirelessNetworks.MobileComputing.KluwerAcademicPublishers,1996,153. [34] Kim,Donghyun,Wu,Yiwei,Li,Yingshu,Zou,Feng,andDu,DingZhu.ConstructingMinimumConnectedDominatingSetswithBoundedDiametersinWirelessNetworks.IEEETransactionsonParallelandDistributedSystems20(2009):147. [35] Law,YeeWei,Hartel,Pieter,Den,JerryHartog,andHavinga,Paul.LinklayerJammingAttacksonSMAC.WirelessSensorNetworks,2005.ProceeedingsoftheSecondEuropeanWorkshopon.2005,217. [36] Levitt,BarryK.FH/MFSKPerformanceinMultitoneJamming.SelectedAreasinCommunications,IEEEJournalon3(1985).5:627. [37] Li,Mingyan,Koutsopoulos,Iordanis,andPoovendran,Radha.OptimalJammingAttacksandNetworkDefensePoliciesinWirelessSensorNetworks.INFOCOM2007.26thIEEEInternationalConferenceonComputerCommunications.IEEE.2007,1307. [38] Li,Yingshu,Kim,Donghyun,Zou,Feng,andDu,DingZhu.ConstructingConnectedDominatingSetswithBoundedDiametersinWirelessNetworks. 139 PAGE 140 [39] Li,Yingshu,Thai,MyT.,Wang,Feng,andDu,DingZhu.OntheConstructionofaStronglyConnectedBroadcastArborescencewithBoundedTransmissionDelay.IEEETransactionsonMobileComputing5(2006).10:1460. [40] Li,Yingshu,Thai,MyT.,Wang,Feng,Yi,ChihWei,Wan,PengJun,andDu,DingZhu.OnGreedyConstructionofConnectedDominatingSetsinWirelessNetworks:ResearchArticles.Wirel.Commun.Mob.Comput.5(2005).8:927. [41] Li,Yingshu,Zhu,Shiwei,Thai,MyT.,andDu,DingZhu.LocalizedConstructionofConnectedDominatingSetinWirelessNetworks.NSFInternationalWorkshoponThoreticalAspectsofWirelessadhoc,SensorandPeertoPeerNetworks.2004. [42] Ling,Qi,Li,Tongtong,andDing,Zhi.ANovelConcept:MessageDrivenFrequencyHopping(MDFH).Communications,2007.ICC'07.IEEEInternationalConferenceon.2007,5496. [43] Ma,Jianqing,Zhong,Yiping,andZhang,Shiyong.FrequencyHoppingBasedSecureSchemesinSensornets.CIT'05:ProceedingsoftheTheFifthInternationalConferenceonComputerandInformationTechnology.Washington,DC,USA:IEEEComputerSociety,2005,459. [44] Martin,Burkhart,Pascal,VonRickenbach,Roger,Wattenhofer,andAaron,Zollinger.DoesTopologyControlReduceInterference?MobiHoc'04:Proceedingsofthe5thACMinternationalsymposiumonMobileadhocnetworkingandcomputing.NewYork,NY,USA:ACM,2004,9. [45] Mohammed,K.,Gewali,L.,andMuthukumar,V.GeneratingQualityDominatingSetsforSensorNetwork.ICCIMA'05:ProceedingsoftheSixthInternationalConferenceonComputationalIntelligenceandMultimediaApplications.Washington,DC,USA:IEEEComputerSociety,2005,204. [46] Mu,Fi,Brim,Lubos,Cerna,Ivana,Krical,Pavel,andPelanek,Radek.DistributedShortestPathforDirectedGraphswithNegativeEdgeLengths.Tech.rep.,MasarykUniversity,2001. [47] Navda,Vishnu,Bohra,Aniruddha,andGanguly,Samrat.UsingChannelHoppingtoIncrease802.11ResiliencetoJammingAttacks.INFOCOM2007.26thIEEEInternationalConferenceonComputerCommunications.IEEE.2007,2526. [48] Niculescu,DragosandLab,Dataman.Adhocpositioningsystem(APS)usingAOA.INFOCOM2003.TwentySecondAnnualJointConferenceoftheIEEEComputerandCommunicationsSocieties.IEEE.vol.3.2003,1734. [49] Niculescu,DragosandNath,Badri.AdHocPositioningSystem(APS).GLOBECOM2001.Nov2001. 140 PAGE 141 O'Rourke,Joseph.ComputationalGeometryinC(CambridgeTractsinTheoreticalComputerScience).CambridgeUniversityPress,1998. [51] Perkins,CharlesandRoyer,Elizabeth.AdhocOnDemandDistanceVectorRouting.InProceedingsofthe2ndIEEEWorkshoponMobileComputingSystemsandApplications.1997,90. [52] Perrig,Adrian,Stankovic,John,andWagner,David.SecurityinWirelessSensorNetworks.Commun.ACM47(2004).6:53. [53] Popper,Christina,Strasser,Mario,andCapkun,Srdjan.JammingResistantBroadcastCommunicationwithoutSharedKeys.ETHZurichDINFKTechnicalReport609.2008. [54] Ruan,Lu,Du,Hongwei,Jia,Xiaohua,Wu,Weili,Li,Yingshu,andKo,KerI.AGreedyApproximationforMinimumConnectedDominatingSets.Theor.Comput.Sci.329(2004).13:325. [55] Sarma,HirenKumarDevaandKar,Avijit.SecurityThreatsinWirelessSensorNetworks.CarnahanConferencesSecurityTechnology,Proceedings200640thAnnualIEEEInternational.2006,243. [56] Sen,Arunabha,Roxborough,Tom,andSinha,BhabaniP.OnanOptimalAlgorithmforChannelAssignmentinCellularNetworks.ICC'99Internationalconferenceoncommunication.1999. [57] Sharma,MohanB.,Mandyam,NarasimhaK.,andIyengar,SitharamaS.AnOptimalDistributedDepthFirstSearchAlgorithm.CSC'89:Proceedingsofthe17thconferenceonACMAnnualComputerScienceConference.NewYork,NY,USA:ACM,1989,287. [58] Sidek,OthmanandYahya,Abid.ReedSolomonCodingforFrequencyHoppingSpreadSpectruminJammingEnvironment.AmmericanJournalofAppliedSciences.vol.5.2008,1281. [59] Sivakumar,Raghupathy,Das,Bevan,andBharghavan,Vaduvur.AnImprovedSpinebasedInfrastructureforRoutinginAdHocNetworks.Proc.ofTheThirdIEEESymposiumonComputersandCommunications(ISCC).1998. [60] Strasser,Mario,Danev,Boris,andCapkun,Srdjan.DetectionofReactiveJamminginSensorNetworks.ETHZurichDINFKTechnicalReport(2009). [61] Strasser,Mario,Popper,Christina,Capkun,Srdjan,andCagalj,Mario.JammingResistantKeyEstablishmentusingUncoordinatedFrequencyHopping.IEEE,2008. [62] Sun,HungMin,Hsu,ShihPu,andChen,ChienMing.MobileJammingAttackanditsCountermeasureinWirelessSensorNetworks.AdvancedInformation PAGE 142 [63] Tague,Patrick,Slater,David,Poovendran,Radha,andNoubir,Guevara.LinearProgrammingModelsforJammingAttacksonNetworkTrafcFlows.ModelingandOptimizationinMobile,AdHoc,andWirelessNetworksandWorkshops,2008.WiOPT2008.6thInternationalSymposiumon.2008,207. [64] Thai,MyT.,Tiwari,Ravi,andDu,DingZhu.OnConstructionofVirtualBackboneinWirelessAdHocNetworkswithUnidirectionalLinks.IEEETransactionsonMobileComputing7(2008).9:1098. [65] Thai,MyT.,Tiwari,Ravi,andDu.,DingZhu.OnConstructionofVirtualBackboneinWirelessAdHocNetworkswithUnidirectionalLinks.IEEETransactionsonMobileComputing7(2008):1098. [66] Thai,MyT.,Wang,Feng,Liu,Dan,Zhu,Shiwei,andDu.,DingZhu.ConnectedDominatingSetsinWirelessNetworkswithDifferentTransmissionRanges.IEEETransactionsonMobileComputing6(2007).7:721. [67] Thai,MyT.,Wang,Feng,Liu,Dan,Zhu,Shiwei,andDu,DingZhu.ConnectedDominatingSetsinWirelessNetworkswithDifferentTransmissionRanges.IEEETransactionsonMobileComputing6(2007).7:721. [68] Thai,MyT.,Zhang,Ning,Tiwari,Ravi,andXu,Xiaochun.OnApproximationAlgorithmsofkconnectedmdominatingSetsinDiskGraphs.Theor.Comput.Sci.385(2007).13:49. [69] Thuente,David,Newlin,Ben,andAcharya,Mithun.JammingVulnerabilitiesof802.11e.Proceedingsofthe26thIEEECommunicationsSocietyMilitaryCommunicationsConference(MILCOM).2007. [70] Torrieri,DonJ.FundamentalLimitationsonRepeaterJammingofFrequencyHoppingCommunications.SelectedAreasinCommunications,IEEEJournalon7(1989).4:569. [71] Toshihiro,Fujito.OnApproximabilityoftheIndependentConnectedEdgeDominatingSetProblems.FSTTCS2000:Proceedingsofthe20thConferenceonFoundationsofSoftwareTechnologyandTheoreticalComputerScience.London,UK:SpringerVerlag,2000,117. [72] Toshihiro.,Fujito.HowtoTrimanMST:A2ApproximationAlgorithmforMinimumCostTreeCover.ICALP1(2006):431. [73] Tseng,YuChee,Ni,SzeYao,Chen,YuhShyan,andSheu,JangPing.TheBroadcastStormProbleminaMobileAdHocNetwork.Wirel.Netw.8(2002).2/3:153. 142 PAGE 143 Wan,PengJun,Alzoubi,M.KhaledM.,andFrieder,Ophir.DistributedConstructionofConnectedDominatingSetinWirelessAdhocNetworks.Mob.Netw.Appl.9(2004).2:141. [75] Wang,Feng,Thai,MyT.,andDu,DingZhu.OntheConstructionof2connectedVirtualBackboneinWirelessNetworks.Trans.Wireless.Comm.8(2009).3:1230. [76] Wang,Hang,Guo,Jingbo,andWang,Zanji.FeasibilityAssessmentofRepeaterJammingTechniqueforDSSS.WirelessCommunicationsandNetworkingConference,2007.WCNC2007.IEEE(2007):2322. [77] Wood,AnthonyD.,Stankovic,JohnA.,andSon,SangH.JAM:aJammedAreaMappingServiceforSensorNetworks.RealTimeSystemsSymposium,2003.RTSS2003.24thIEEE.2003,286. [78] Wood,AnthonyD.,Stankovic,JohnA.,andZhou,Gang.DEEJAM:DefeatingEnergyEfcientJamminginIEEE802.15.4basedWirelessNetworks.Sensor,MeshandAdHocCommunicationsandNetworks,2007.SECON'07.4thAnnualIEEECommunicationsSocietyConferenceon.2007,60. [79] Wu,JieandLi,Hailan.OnCalculatingConnectedDominatingSetforEfcientRoutinginAdHocWirelessNetworks.DIALM'99:Proceedingsofthe3rdinternationalworkshoponDiscretealgorithmsandmethodsformobilecomputingandcommunications.NewYork,NY,USA:ACM,1999,7. [80] Wu.,YiweiandLi,Yingshu.ConstructionAlgorithmsforkconnectedmdominatingSetsinWirelessSensorNetworks.MobiHoc'08:Proceedingsofthe9thACMinternationalsymposiumonMobileadhocnetworkingandcomputing.NewYork,NY,USA:ACM,2008,83. [81] Wu,YiweiandLi,Yingshu.ConstructionAlgorithmsforkconnectedmdominatingSetsinWirelessSensorNetworks.MobiHoc'08:Proceedingsofthe9thACMinternationalsymposiumonMobileadhocnetworkingandcomputing.NewYork,NY,USA:ACM,2008,83. [82] Wu,Yiwei,Wang,Feng,Thai,MyT.,andLi,Yingshu.ConstructinkConnectedmDominatingSetsInWirelessSensorNetworks.October,2007. [83] Xu,Wenyuan.ChannelSurng:DefendingWirelessSensorNetworksfromInterference.InformationProcessinginSensorNetworks,2007.IPSN2007.6thInternationalSymposiumon.2007,499. [84] Xu,Wenyuan,Ma,Ke,Trappe,Wade,andZhang,Yanyong.JammingSensorNetworks:AttackandDefenseStrategies.Network,IEEE20(2006).3:41. 143 PAGE 144 Xu,Wenyuan,Trappe,Wade,andZhang,Yanyong.AntijammingTimingChannelsforWirelessNetworks.WiSec'08:ProceedingsoftherstACMconferenceonWirelessnetworksecurity.NewYork,NY,USA:ACM,2008,203. [86] Xu,Wenyuan,Trappe,Wade,Zhang,Yanyong,andWood,Timothy.TheFeasibilityofLaunchingandDetectingJammingAttacksinWirelessNetworks.MobiHoc'05:Proceedingsofthe6thACMinternationalsymposiumonMobileadhocnetworkingandcomputing.NewYork,NY,USA:ACMPress,2005,46. [87] Xu,Wenyuan,Wood,Timothy,Trappe,Wade,andZhang,Yanyong.ChannelSurngandSpatialRetreats:DefensesagainstWirelessDenialofService.WiSe'04:Proceedingsofthe3rdACMworkshoponWirelesssecurity.NewYork,NY,USA:ACMPress,2004,80. [88] Zhang,Ning,Shin,Incheol,Zou,Feng,Wu,Weili,andThai,MyT.TradeoffSchemeforFaultTolerantConnectedDominatingSetsonSizeandDiameter.FOWANC'08:Proceedingofthe1stACMinternationalworkshoponFoundationsofwirelessadhocandsensornetworkingandcomputing.NewYork,NY,USA:ACM,2008,1. [89] Zhang,YifengandDill,Jeffrey.AnAntijammingAlgorithmusingWaveletPacketModulatedSpreadSpectrum.MilitaryCommunicationsConferenceProceedings,1999.MILCOM1999.IEEE.vol.2.1999,846vol.2. 144 PAGE 145 IncheolShinwasbornin1977,inSeoul,RepublicofKorea.HereceivedBachelorofengineeringdegreeatcomputerengineeringin2002fromHansungUniversity,Seoul,RepublicofKorea.In2006,hereceivedhisMasterofEngineeringdegreefromtheDepartmentofComputerandInformationScienceandEngineeringattheUniversityofFlorida.Hismajorresearchareaiscomputernetworks. 145 