Citation
Data distribution and algorithms for asynchronous parallel processing of object-oriented knowledge bases

Material Information

Title:
Data distribution and algorithms for asynchronous parallel processing of object-oriented knowledge bases
Creator:
Thakore, Arun Kumar, 1962- ( Dissertant )
Su, Stanley Y. W. ( Thesis advisor )
Taylor, Fred ( Reviewer )
Navathe, Shamkant B. ( Reviewer )
Chow, Yuan-Chieh ( Reviewer )
Lam, Herman ( Reviewer )
Phillips, Winfred M. ( Degree grantor )
Lockhart, Madelyn M. ( Degree grantor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1990
Language:
English
Physical Description:
ix, 302 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Connectivity ( jstor )
Correlations ( jstor )
Data processing ( jstor )
Data transmission ( jstor )
Databases ( jstor )
Identifiers ( jstor )
Students ( jstor )
Teacher organizations ( jstor )
Teachers ( jstor )
Vertices ( jstor )
Algorithms ( lcsh )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Object-oriented databases ( lcsh )
Parallel processing (Electronic computers) ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Sophisticated management and reasoning about large quantities of complex data are essential in advanced application areas. Several Object-Oriented (OO) databases/knowledge bases have been developed to effectively capture the complex domain knowledge. However, due to the enormity and intricacy of the data, and the generality of the functions implemented by the OO databases/knowledge bases, the existing implementations operate inefficiently. In this dissertation we study several issues related to the efficient parallel implementation of OO knowledge bases. The physical organization of the data across the processing nodes of a parallel system plays an important role in determining the execution time. We present several techniques for efficiently partitioning large quantities of OO data across the processing nodes of the parallel system. The techniques take advantage of the structure and the semantic propertyof the OO data in localizing manipulation and reducing the overall communication costs during query processing. Further, we present parallel algorithms for the processing of non-deductive and deductive queries against a large OO knowledge base. The algorithms are developed for various query complexities. During processing, the algorithms avoid the execution of time-consuming join operations by retrieving the explicitly stored relationships, among the various object instances, based on patterns of object associations. Generation of large quantities of temporary data is avoided by marking object instances using their identifiers and by employing a two-phase query processing strategy. A query is process by concurrent multiple wavefronts, thereby improving parallelism and avoiding the complexities introduced in their sequential implementation. the suitability of the data partitioning techniques and the correctness of the performance of the parallel algorithms have been tested and analyzed by running parallel programs on the IBM's distributed message passing system Victor. Benchmark queries of different semantic complexities are generated and their performance is analyzed for various data and system parameters. The performance of several application domains characterized by specific mixes of the benchmark queries is also analyzed.
Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 293-301).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Arun Kumar Thakore.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Arun Kumar Thakore. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
026075936 ( ALEPH )
24887658 ( OCLC )
AHY9416 ( NOTIS )

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Full Text













DATA DISTRIBUTION AND ALGORITH14S FOR ASYNCHRONOUS PARALLEL
PROCESSING OF OBJECT-ORIENTED KNOWLEDGE BASES
















By

ARUN KUMAR THAKORE


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


1990

..












































Copyright 1990

by

Arun Kumar Thakore

..






























To my wife and parents

..












ACKNOWLEDGEMENTS


I take this opportunity to express my deepest gratitude to Dr. Stanley Y.W. Su. He has been and is a constant source of inspiration to me. He has motivated me and guided me with

utmost patience. I have had dif ficult times during the course of my PhD work. He has supported me and understood my

frustrations, and has always been a trusted mentor. I also

thank Dr. Shamkanth Navathe for his encouragement and support. He has always given me timely suggestions and made me feel at home.

I thank Dr. Herman Lam for his helpful suggestions. His acumen for details has helped me in improving upon my work. I also thank Dr. Fred Taylor and Dr. Randy Chow for being on my committee. My appreciation and admiration goes to Sharon

Grant whose serene face and charming smile brighten each working day. She seems tireless and is always ready to help with a smiling patience even at the end of a hectic day. I

thank my friends at the Database Research and Development Center for their enthusiasm and cooperation.

This work was supported by a grant from the National Science Foundation and the Florida High Technology Council. The IBM Research Center at Yorktown Heights provided the use

..





of their facilities for the simulation of the ideas developed

in this research. The support of National Science Foundation, Florida High Technology Council, and IBM is greatly acknowledged.

I am thankful to my wife Rina who has provided me with

incredible understanding and encouragement in innumerous ways. Her patience and moral support were essential in the

completion of this work. Last but not the least, I am

eternally grateful to my parents for their love and encouragement in all my endeavors.

..










TABLE OF CONTENTS


ACKNOWLEDGEMENTS. iv



CHAPTERS

1 INTIRODUJCTION. 1

2 SURVEY OF REL.ATED WORK o 9

Database Machines . 9 Knowledge Base Machines. 16
Logic Based Machines . . . 16
Production System Machines .2
Semantic Network Machines . . 25

3 REPRESENTATION AND, QUERYING OF
OBJECT-ORIENTED DATABASES .o. 29

object-oriented View of Databases 30
A Closed Model of Query Processing for
object-oriented Databases .*.34 Association Operator . . . .35 NonAssociation Operator. . . .36
Query Examples 36
Noncyclic Association Pattern 37
Cyclic Association Pattern .4
Deductive Queries . . . 42

4 PARALLEL ARCHITECTURAL MODEL AND
DATA ORGANIZATION. . . . . . o . . .47

Parallel Architectural Model . o . .47 Partitioning and Mapping of Data 50
Data Clustering . .52 Load Balancing. . . . . o . .59
Mapping of Cluster Groups Onto Processors 72

..






5 PARALLEL ALGORITHMS FOR NON-DEDUCTIVE
QUERY PROCESSING 106

Processing Phases 106 Parallel Algorithms 110
Identification of Subdatabases 112 Generation of the Result 136

6 PARALLEL ALGORITHMS FOR PROCESSING OF
DEDUCTIVE RULES .175

Processing Phases . 175 Parallel Algorithms. 178
Derivation of the Target Subdatabase. 180 Processing of Linearly Recursive Rules 187

7 SIMULATION ENVIRONMENT AND RESULTS.212

Simulation Environment. 213
Hardware. . 213 Software Components. .214 Benchmark Queries. 217 Database Characteristics. .219
Simulation Results and Analysis. 220
Suitability of the Heuristic Mapping
Techniques. 220
Effect of Data and Query Parameters on
Performance.226
Effect of System Parameters on Performance 239
Effect of Derivation Parameters on
Performance .248

8 CONCLUSION 281

APPENDICES

A EQUATIONS CHARACTERIZING DATA PARAMETERS 285

B EQUATIONS CHARACTERIZING SIMULATED TIMINGS 289

REFERENCES 293

BIOGRAPHICAL SKETCH .302


vii

..













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

DATA DISTRIBUTION AND ALGORITHMS FOR ASYNCHRONOUS PARALLEL PROCESSING OF OBJECT-ORIENTED KNOWLEDGE BASES By

ARUN KUMAR THAKORE

DECEMBER 1990

Chairman: Dr. Stanley Y. W. Su Major Department: Electrical Engineering

Sophisticated management and reasoning about large quantities of complex data are essential in advanced application areas. Several Object-Oriented (00)

databases/knowledge bases have been developed to effectively

capture the complex domain knowledge. However, due to the enormity and the intricacy of the data, and the generality of

the functions implemented by the 00 databases/knowl edge bases, the existing implementations operate inefficiently. In this

dissertation, we study several issues related to the efficient parallel implementation of 00 knowledge bases.

The physical organization of the data across the processing nodes of a parallel system plays an important role in determining the execution time. We present several

techniques for efficiently partitioning large quantities of 00 data across the processing nodes of the parallel system. The


viii

..







techniques take advantage of the structure and the semantic property of the 00 data in localizing manipulation and reducing the overall communication costs during query processing.

Further, we present parallel algorithms for the processing of non-deductive and deductive queries against a

large 00 knowledge base. The algorithms are developed fQr various query complexities. During processing, the algorithms avoid the execution of time-consuming join operations by retrieving the explicitly stored relationships, among the various object instances, based on patterns of object associations. Generation of large quantities of temporary data is avoided by marking object instances using their

identifiers and by employing a two-phase query processing strategy. A query is processed by concurrent multiple

wavefronts., thereby improving parallelism and avoiding the complexities introduced in their sequential implementation.

The suitability of the data partitioning techniques and

the correctness and the performance of the parallel algorithms have been tested and analyzed by running parallel programs on the IBM's distributed message passing system Victor. Benchmark queries of different semantic complexities are generated and their performance is analyzed for various data

and system parameters. The performance of several application domains characterized by specific mixes of the benchmark queries is also analyzed.

..











CHAPTER 1
INTRODUCTION


Many advanced database application areas such as CAD/CAM, CASE, and decision support have an increasing need for manipulating large quantities of data having complex structures. Relational systems are not expressive enough to capture the complex structural relationships and the behavioral properties of objects found in more advanced applications. Several Object-Oriented (00) Semantic data models have been developed [HAM81, BAT85, HUL87, SU89] based on the features of the popular Object-Oriented programming paradigm. These models provide a variety of constructs to effectively model complex domain knowledge. Several 00

database systems have been implemented [FIS87, KIN84, WOE86]. The generality and expressiveness introduced by the 00 models make it easier for the user to model large quantities of application data in a complex domain. However, the need for querying and reasoning about a large number of complex data objects and relationships among them causes the existing 00 systems to operate rather inefficiently.

A number of database machines have been proposed to improve the query processing efficiency of large databases. Also, researchers in the Artificial Intelligence (AI) area

1

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2

have designed hardware architectures and processing techniques for efficiently supporting the various reasoning mechanisms encountered in the execution of expert systems. However, as

illustrated in this chapter and elaborated further in the next chapter, they do not provide adequate means to satisfy the efficient manipulation and reasoning needs of large complex 00 knowledge bases.

The database machines have used a variety of techniques to overcome the 1/0 and processor-memory bottlenecks of the Von Neumann architecture. They have primarily supported the

efficient execution of the time consuming primitive relational operations, such as, join and set processing, on large

relational databases. The'requirements and characteristics of 00 systems are different from those of the relational systems. Hence, the architectures and the algorithms for the efficient execution of 00 systems can be expected to be different from those proposed for relational systems.

For example, the domain knowledge in an 00 database can be represented by objects and their associations. The query

languages used for querying the database should be patternbased (ALA89a, ALA89bJ as opposed to the attribute-based relational languages. Pattern-based languages allow the user to express the query as complex patterns specifying the object classes, their associations with other object classes,

relationships desired, and the operation(s) to be performed on the selected objects. Using relational machines, processing

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3

of 00 queries will involve the execution of time consuming join operations. This is due to the fact that in a relational model the interrelationships among data objects are scattered across several relations and have to be recomputed during query processing. The frequency of the join operations grows as the complexity of the data objects and the interrelationships among the data objects increase. Obviously, applications with dense interrelated complex objects require the modeling power of an 00 semantic model. Further, the relational database machines provide efficient support only for queries involving retrieval and storage of large quantities of data. Processing of deductive queries involving manipulation of large sets of deductive rules and factual data is not supported by these database machines.

Recently, Bic and Hartman have proposed an Active Graph Model (AGM) for database processing[BIC89]. The AGM is

proposed for improving the efficiency and parallelism during query processing. The AGM explicitly captures the

relationships among the data elements and processes the query

by injecting tokens from various data elements and propagating them asynchronously along the relationship arcs. The explicitly captured relationships eliminate the need for computing them during processing by executing time consuming join operations, thereby, improving the query processing efficiency. The asynchronous nature of the processing improves parallelism by eliminating the need for centralized

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4

control at every execution step. However, the granularity of computation and the query model of the AGM are not suitable for the efficient processing of large 00 systems.

The granularity of computation in the AGM is at the data

element level. In 00 systems, the number of data objects, the connectivity among the data objects, and the number of bytes describing the properties of each object can be very large. At low granularities of computation, this can lead to

generation and processing of an excessive number of tokens carrying a substantial amount of information with them. This in turn can lead to a significant increase in overhead costs. Also, using the AGM, the query is issued against a database which is represented as a network of interrelated data elements. However, the result of the query is a normalized

relation, where different tuples of the relation are collected from the selected nodes of the target set. Since the result of the query is not structurally represented in the same form as the original database, the result of the query cannot be

stored and further uniformly operated on by the same query model to produce other results that satisfy other qualification conditions. Thus, the closure property is not maintained.

AI machines have provided efficient reasoning of expert systems by implementing in hardware the data structures and operations on the data structures used during the reasoning process. Expert systems are used in narrow domains and are

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5

associated with relatively small sets of facts and rules. The AI architectures and processing techniques employed by them assume that the fact and rule bases are main memory resident. The hardware and software techniques used by the AI machines cannot be efficiently applied for reasoning on large

quantities of complex data and rules stored across several secondary storage devices.

In this dissertation, we present and experimentally analyze several techniques for efficiently partitioning and

processing large 00 knowledge bases on parallel architectures. The data partitioning heuristics and nondeductive and deductive query processing algorithms developed in this work

are general and can be executed on a variety of parallel machines. The main features of the proposed techniques are as follows:

1) Similar to AGM, the interrelationships among the data objects are explicitly stored and used during query processing. This eliminates the need for the execution of time consuming join and unification operations in order to relate data objects during the processing of queries referencing the relationships captured by the 00 model. 2) The techniques take advantage of the structure and the semantic property of the 00 data in localizing manipulation and reducing the overall communication costs during processing.

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6

3) Unlike AGM, the query processing techniques are based on

an 00 query model which maintains the closure property. Thus, the result of a query is structured and represented in an 00 framework similar to the base data.

4) Similar to AGM', an asynchronous approach is adopted in the processing of queries. However, the granularity of processing is at a object class level rather than at the data element level. This enables the exploitation of parallelism without

the overhead penalties associated with the processing of a large number of tokens. Further, data blocks within the object class are pipelined and temporal parallelism is exploited in the processing.

5) A two-phase processing. strategy has been used to eliminate the unnecessary generation and movement of large quantities of descriptive data. During the first phase of the processing,

all the objects in the database satisfying the query are marked after manipulating the associative data. Subsequently, the selected descriptive data of only the marked objects are retrieved and presented to the user. 6) A user query is processed by multiple concurrent wavefronts. Each wavefront is asynchronously executed by a

pipeline of relevant processors. Desired objects of various classes and specified relationships among the selected objects are stored in a distributed fashion as a result of the

processing. This is in contrast to the traditional rigid tree structured control in the processing of relational queries.

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7

This improves the overall parallelism in the processing and eliminates the complexities involved in the sequential implementation. 1

7) The derivation rules of the knowledge base system are integrated into the 00 data based on the class(es) of objects and/or new relationships among the classes of objects they derive. The integrated structure facilitates in focusing on

the desired set of data and rules from the large knowledge base during processing.

8) During the derivation process, the various classes of objects and/or the various relationships among the specified

classes of objects are derived in parallel by various assigned processing nodes. Further, different rules deriving the objects of a class or the relationships between two classes

are executed in parallel. This strategy increases the overall parallelism in the processing and a distributed controlling mechanism is implemented.

This dissertation is organized as follows. In Chapter 2, we survey the related work on architectures and techniques for improving the performance of databases and knowledge bases. In Chapter 3, we present the 00 view of knowledge bases and discuss the features of an 00 query language based on an 00 query model which maintains the closure property. In Chapter 4, we describe a parallel architectural model for the

implementation of large 00 knowledge bases and present several heuristic techniques for the efficient mapping of the 00 data

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8

across the nodes of the parallel architecture. In Chapter 5, we present asynchronous parallel algorithms for processing nondeductive queries against 00 knowledge bases. The

algorithms are developed for various complexities of the queries. Further, in Chapter 6,, we present algorithms and a distributed control mechanism for the parallel processing of deductive queries against 00 knowledge bases. The

effectiveness of various heuristic data mapping techniques, and the correctness of the parallel algorithms and their performance is studied by implementing the algorithms and the controlling mechanism on a parallel message passing system.

The results of the simulation are presented in Chapter 7. Finally, our conclusions and possible future research directions are presented in Chapter 8.

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CHAPTER 2
SURVEY OF RELATED WORK


In this chapter, we will survey the related work in the areas of database machines and knowledge base machines. Database machines have been developed to improve the processing efficiency of large databases. Similarly, knowledge base machines provide efficient means of reasoning on data.

2.1. Database Machines


Since the advent of YLSI technology and reduction in hardware costs, there has been a trend in the use of multicomputer systems for database applications. Multicomputer systems obtain considerable performance improvement over von Neumann architecture by decomposing the computational task into a number of parallel subtasks and executing them simultaneously on different processors. These architectures employ several techniques in order to improve the utilization of hardware resources and to reduce the query execution time of large databases. In this section, we will survey some of the recent multicomputer database systems that are relevant to the research presented in this dissertation. The goal of the survey is to illustrate the well established

9

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10
techniques that can be adopted in our research and also to depict the limitations of the existing work.

The database initially resides on Secondary Storage Devices (SSDs). At the onset of the processing large quantities of data are moved into main memory and during processing the temporary results are staged in and out of secondary storage. The time for I/O is a major source of inefficiency in database processing. Database machines [see references in HSI83, OZK86, SU88] employ multiple processors each with their own main memory and SSDs. The data files are partitioned into subfiles and stored in a distributed fashion across the SSDs. During processing, different subfiles are loaded into main memory simultaneously from various secondary storage devices. The parallel retrieval of data relieves the I/O bottleneck. Architectures such as GRACE [KIT84] also employ filter processors integrated into the disk modules. The filter processor performs the selection and projection on the fly and reduces the amount of unnecessary data staged into the main memory for further processing.

The organization and distribution of data across the SSDs also plays an important role in reducing the I/O costs. SM3, DIRECT, and Cube-Connected Multiprocessor [BARU86, DEW79, FRI87] horizontally partition the data files into equal segments and distribute the segments across the SSDs. Such a distribution balances the retrieval in addition to improving the retrieval parallelism. However, since the characteristics

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of the data are not known, all the data partitions have to be retrieved and processed. GAMMA, DBC/1012, and GRACE hash the tuples of the relations into various partitions based on the hash values of some selected attributes [DEW86, TER84, KIT84]. These partitions are equally distributed among the available SSDs. During query processing, data retrieval can be reduced by retrieving only the relevant partitions and by ignoring partitions whose hashed values do not satisfy those desired by the query. GRACE sorts the tuples within each partition in addition to hashing the tuples into partitions. Sorting

eliminates the need to compare all the data values during processing. Hashing requires additional processing overheads. Moreover, efficient processing of queries involving nonhashed attribute values cannot be guaranteed.

DBC and MDBS process the data based on an attribute-based model and data records are clustered based on the semantic similarity of their contents [BAN89]. Records of a cluster are evenly distributed across the SSDs of various computers. The clusters of records that are relevant to a search query can be quickly located and retrieved from disks. However, queries containing search conditions that do not match the predefined descriptors on which the clusters are based do not have the same efficiency as those that do. Data files are vertically partitioned by the DSM and the OFC architectures [COP85, LEE89]. In the DSM, a relation is fully decomposed into binary relations. Each binary relation contains the

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surrogates and the values of an individual attribute of the original relation. However, the OFC vertically partitions the relations based on the associative and the descriptive data. Vertical partitioning reduces the amount of data retrieved by retrieving only the partition(s) containing the values of the attribute(s) referenced by the query. However, update costs are higher for vertically partitioned data.

Processing of the join operation is very time consuming. It involves relating data between two distributed relations. In addition to retrieval of data, sizable data may have to be exchanged among the processors. Database machines employ a variety of techniques to improve the performance of the join operation. SM3 and the Cube-Connected Multiprocessor use nested-loop join algorithm wherein the smaller relation is transmitted among the processors and joined with all the horizontal segments of the larger relation [BARU86, FRI87]. SM3 reduces data transfer time by using a memory switching scheme, whereas Cube-Connected Multiprocessor takes advantage of the increased connectivity among the processors. DBC/1012, GAMMA, and GRACE use hash-based join algorithm to reduce the amount of data transferred among the processors [TER84, DEW86, KIT84]. Valduriez [VAL87] has proposed prejoining the relations based on the primary keys and storing the join indices as prejoined relations. This considerably improves the performance of the join operation. Similar technique has been used by the OFC [LEE89].

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Processing of complex queries involves execution of a sequence of a large number of join operations. Although

various database machines use several techniques to improve

the performance of the individual join operations, similar performance improvements cannot be expected for overall query execution. The join algorithms take advantage of the even distribution of data. However, even distribution of data at the end of the operation cannot be guaranteed. Processing of

subsequent join operations in the query may be inefficient due to the unevenly distributed data. In addition to poor

performance, low hardware utilization can be expected due to uneven computational loads on the processors.

In the Cube-Connected Multiprocessor, redistribution of

the result data is suggested and redistribution algorithms have been designed [FRI87J. Redistribution of the result data after every operation may improve the performance of subsequent operations. However, it can itself be very time

consuming. In DIRECT multiprocessor [DEW89J, Query Processors (QPs) are assigned to process individual operations of the query tree. A QP starts execution when data are available at

each of its input node(s) and the result is transferred to the QP processing the subsequent operation. Moreover, DIRECT is a MIMD machine and multiple queries are processed at the same time. The data flow approach and the MIND nature of the processing enable DIRECT to improve its resource utilization

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and query execution time. A data flow approach is also used by GAMMA, GRACE, and OFC [DEW86, KIT84, LEE89].

Recently, the need for processing data based on a data model that explicitly captures the semantic relationships among the data has been established [BIC86, BIC89, LEE89]. It is observed that the relational model scatters the relationships across several relations and during processing the desired relationships have to be computed by performing time consuming join operations. The processing of the OFC [LEE89] is based on an Object-Oriented Semantic Model. OFC captures the relationships and the descriptive data about the objects of various object classes in the form of unnormalized or generalized relations. A number of primitive database operations on the generalized relations have been identified in the OFC. Similar to the relational approach, a query is compiled into a tree of primitive operations. Efficiency is obtained by replacing the join operations of the relational model with efficient special join operations which take advantage of the explicitly captured relationships. Further, in order to reduce the amount of unnecessary data transferred among the processors, a two-phase processing strategy is employed. During the first phase, a skeletal nonnormalized relation of object identifiers is formed. Subsequently, the desired descriptive data of only the identifiers in the resulting relation are retrieved. However, OFC takes a

relational approach in the processing of semantic data and

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does not eliminate the processing of time consuming join operations.

Similarly, AGM represents the database as a network of

interrelated entities and relationships [BIC86, BIC89]. A query is represented as a directed tree of interrelated data sets. The desired restrictions are also specified in the query. The query is processed by injecting tokens from various data elements and propagating them asynchronously along the arcs of the network. The tokens carry the status of the selection conditions as well as desired descriptive

values. Unlike the OFC, the network representation allows the AGM to eliminate the processing of join operations. In

addition, the asynchronous nature of the processing improves the processing parallelism. However, the granularity of the

computation is at the data element level and a large number of tokens carrying a substantial amount of data have to be generated, transmitted, and processed. This can significantly increase the overhead costs.

In the research presented in this dissertation, similar to the AGM, we represent the Object-oriented (00) data as a

network of interrelated objects and adopt an asynchronous model of computation. However, the granularity of computation is higher and we cluster the objects and relationships of various classes and manipulate them similarly. A higher

granularity enables in reducing the overhead costs. We employ a two-phase processing strategy similar to the OFC in order to

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reduce the amount of unnecessary data transferred among the processors. Unlike the OFC and the AGM, the query processing in our research is based on an 00 query model that maintains the closure property. The result of the query is represented in the similar network form as the input to the query. This enables the output of a query to be further processed using the operators of the same query model. Unlike database

machines, our research deals with efficient query processing

strategies which include the processing of large quantities of deductive rules integrated with a large factual database in an 00 framework.

2.2. Knowledge Base Machines



In this section, we w ill survey the past and the current efforts in designing architectures for knowledge based systems. The architectures can be classified based on the underlying knowledge representation scheme they support. We

will survey the architectures under the following categories:

(1) Logic based machines, (2) Production System machines, and

(3) Semantic Network machines.

2.2.1. Logic Based Machines

These architectures are designed to process knowledge represented in logical statements efficiently. Using this representation scheme, the domain knowledge about objects and their inter-relationships is represented as declarative clauses. There are two kinds of clauses: facts and rules.

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The facts capture specific knowledge that is known to be true, whereas the rules capture general knowledge and can be used in conjunction with facts in deducing information while answering users queries. Prolog is a programming language that is based on logic. Various sequential and parallel

variants of this language have been used as the basis for architectures in this category.

Various architectures have been designed to support the inferencing mechanism of the logic programming system directly in hardware. Architectures [MOR89, TIC88, TAK84, TAM84] are uniprocessors that have been developed to support the

depth-first search strategy and the backtracking mechanism in hardware. Different sources of parallelism in the execution of logic programs have been studied and used in the design of parallel architectures. The various sources of parallelism are as follows:

(i) OR-parallelism--the parallelism in the unification and the simultaneous execution of the various clauses that are unifiable with the given goal clause,

(ii) AND-parallelism--the parallelism in the execution of the sub-goals of the selected clause, (iii) Search-parallelism--the parallelism in the simultaneous search of the sets of clauses that unify with a given goal,

(iv) Unification-parallelism--the parallelism corresponding to the parallel activities within the unification process. Architectures (BEN89, DES88, HER88, IT087, MOT84, SIN89] use

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multiprocessor organizations and a breadth-first search

strategy in exploiting the various sources of parallelism mentioned above. Some of these architectures have used

heuristics in order to guide the search of the inference procedure.

one approach in the design of knowledge based systems has been to combine a relational database system with a logic programming system [K1Y87J. Facts are stored as relations

and managed by the database system, and an inference processor is designed to store rules and perform the reasoning. The

PRISM project at the University of Maryland involves research on a multi-processor configuration knowledge base machine consisting of problem solving machines and database machines [K0H88]. The search and problem solving tasks are handled by the problem solving machines, whereas the database machine performs the unification and database retrieval. The system exploits AND-parallelism, OR-parallelism, and Searchparallelism. Inclusion of a constraint solving machine in the overall architecture is also being considered. The constraint solving machine is a specialized hardware driven by the

problem solving machine and assists in the use of constraints to prune the search space. Similarly, one of the projects at

ICOT in Japan involves combining a relational database machine DELTA and an inference processor PSI over a local area network in order to develop a knowledge base machine (MUR84, WAD87). The DELTA database machine is developed as a

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dedicated hardware and various primitive database operations are implemented in hardware. PSI directly implements the inference mechanism in hardware. It converts the query based on the set of rules into a relational algebraic query which is then manipulated by DELTA. This approach enables the efficient reasoning of large databases and adds deductive capabilities to an existing database system. Nevertheless, this approach is not suitable when the number of rules become large and hence have to be stored in the secondary storage. Moreover, since the two systems are loosely coupled,

inefficiencies crop up due to the interface between them. Since a low-level logical interface exists between the two systems, a large number of commands and responses have to be transferred over the medium connecting the two systems,

thereby reducing the overall performance.

Recently, an integrated approach is being taken in the development of architectures for knowledge based systems consisting of a large rule base and a large fact base [QAD87, WON89]. Using the integrated approach, both facts and rules are stored and managed uniformly. In the Opale machine

[SAB87], a top-dowm evaluation strategy is chosen. In order to reduce the number of disk accesses, a set-oriented approach is taken in the processing. Using this approach, a clause is verified by pipelininig sets of solutions from one process to another, each process verifying the binding in a literal. The chosen strategy allows the exploitation of

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OR-parallelism, Search-parallelism, and the pipelining of AND-processes. A unique feature of this architecture is that it executes unification of sets of goals with clause headers read from the disk "on-the-fly".

The Relational Knowledge Base machine [MON88, MORI86, SAK87, YOK86b, YOK86a] integrates the facts and the rules by developing a relational knowledge model and by providing a hardware architecture to support the processing based on that model. The relational knowledge base model is an enhancement of the relational data model and contains terms consisting of constants, variables, and functions as basic data elements. The relational algebra operations are also enhanced to include the unification operation. A top-down evaluation strategy is chosen in this architecture. The main features of this architecture are (1) use of multiple disk systems to store and retrieve the term relations in a distributed

fashion, (2) use of specialized hardware called the

unification engines for performing the unification

operations, (3) use of multi-ported page memories to reduce the I/O bottleneck, and (4) the use of a clustering technique to filter the irrelevant data. This architecture exploits the OR-parallelism, Search-parallelism, and the Unification-parallelism. The approach taken by this architecture can become inefficient when the set of terms representing facts is large because of the top-down evaluation strategy.

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An integrated knowledge base machine architecture for supporting large sets of rules and facts has been proposed by Shin and Berra [SHI87). Surrogate files are constructed by hashing transformation of terms representing the facts and

the heads of the rule clauses. The surrogate files are distributed across multiple disks. The clause bodies are stored in a separate database. A top-down evaluation

strategy is chosen and the query is evaluated by performing

unification operations on the surrogate files and later binding the selected body clauses. A specialized associative processor for performing the unification on the surrogate files is proposed. An overall tightly coupled shared memory system is proposed for the execution. OR-parallelism and Unification-parallelism ar6 exploited by the system and the execution follows a breadth-first search strategy.

Although logic provides a declarative representation of

knowledge and a powerful database search facility, and has been used in developing many knowledge based systems, it is not without its drawbacks. Logic enforces a rigid control structure and procedural knowledge cannot be efficiently

represented and manipulated. Moreover, invariably relational database systems or normalized tables have been used as the

structure for representing facts. The data pertaining to complex objects and the associations among them is not

modelled explicitly and has to be computed by performing time consuming unification joins.

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2.2.2. Production System Machines

Production System is another form of representing and manipulating knowledge, and is used extensively in the

construction of knowledge-based expert systems. A production system consists of a set of condition-action rules called the production memory, and a set of facts called the working memory. OPS5 is the most often used production system

language. OPS5 employs a forward chaining reasoning strategy and performs a three-phase cyclic operation. The three phases are match, conflict resolution, and act. Measurements on

various production systems have shown that the match phase takes about 90% of the computation time. Forgy developed a

fast sequential matching algorithm called rete [FOR82), in order to speed up the matching phase. Rete algorithm has been modified and various multiprocessor architectures based on the modified rete algorithm have been proposed and

analyzed for executing production systems.

One approach in designing architectures for supporting

production systems has been the use of massively parallel structures. The DADO [STL086, STL87J and NON-VON [SI{A85J are massively parallel architectures consisting of thousands of

Processing Elements (PEs) interconnected to form a complete binary tree. The NON-VON was initially developed for

efficient processing of relational database operations and has been improved to support knowledge processing as well. The DADO architecture has been modelled after NON-VON and shares

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some architectural features. In NON-VON the granularity of the PE is small and it executes instructions broadcast by a control processor synchronously with other PEs. The PEs of the DADO machine are capable of executing in either SIMD or MIMD mode. In the MIMD mode, each PE executes instructions in its own local RAM, independent of other PEs. Speedup mainly results from storing the fact base and the rule base in a distributed fashion, and by associatively matching and updating in parallel. The main disadvantage of these

architectures is the poor utilization of their hardware since only a small percentage of productions get affected in each cycle. Various algorithms have been proposed which attempt to improve the utilization of these architectures.

A coarse grain approach has been taken by the MANJI [MIY87] and the PSM [GUP86, FOR86], and other architectures [ACH89, BUT88]. The production rules are precompiled into a modified version of the Rete network [FOR82]. The state of the fact base is saved in various nodes of the network. The network represents a data flow graph and the nodes are evaluated based on the arrival of data tokens. The MANJI is a special shared memory architecture consisting of tens of powerful Processing Units (PUs) connected by a simple bus. The various nodes of the network are statically mapped to different PUs to obtain maximal parallelism possible and are evaluated dynamically in the order of token arrival. The shared memory has been designed and structured so as to

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eliminate reading contention on the bus and to reduce reading/writing conflicts while accessing the shared memory. The PSM is a simple shared memory architecture. The nodes of the rete network are dynamically assigned to different

processors by a scheduler depending on the availability of the processors. In the approaches taken by these

architectures, due to the precompilation, the dynamic

addition/deletion of rules is difficult to implement.

Recently, multiprocessor architectures have been designed and analyzed, for executing production systems, based on the concept of pipelining. Researchers at the University of Waterloo have developed a parallel model of processing which exploits the inherent parallelism in the rete algorithm in the match phase, in addition to providing a degree of control over the parallelism available in the conflict resolution and act phases [OSH87]. A multiprocessor architecture called MAPPS consisting of homogeneous processing elements connected in a heterogeneous topology has been designed after studying the communication requirements imposed by the model. Thus, the architecture executes the parallel model on a three stage processor pipeline and is tuned towards the efficient processing of a large number of changes in the working memory in each production system cycle. Another pipeline architecture is being developed at the University of Kaiserlautern in West Germany [SCH87]. The overall

architecture is a pipeline of special purpose processing

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elements with distributed memory and control. The processing has been based on the rete algorithm. A special instruction

set has been developed and the processing elements are designed to execute the instruction set efficiently. This increases the overall performance of the architecture.

Production systems are very much like logic based systems with an additional capability of dealing with uncertainty and explanation facility. The problem of rigid control structure and low-level data representation is also present in production systems. In addition, the factual data are redundantly stored across various nodes of the rete network

and the approach taken by the architectures might not be efficient and effective when the knowledge base grows.

2.2.3. Semantic Network Machines

Semantic Network is another popular form of representing knowledge. The declarative knowledge about objects and their interrelationships is represented in the form of a directed graph. The nodes of the graph model concepts, data items or

objects, whereas the interrelationships among the objects are modelled as links interconnecting the nodes. Knowledge

processing involves matching a query graph against the data network. The matching will be either simple or involve using general rules of inference. Various parallel architectures

have been designed and simulated in order to speed up the processing of semantic networks [B1C85, FAH83, SAV67).

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A highly parallel SIMD machine, called the Connection Machine [HIL85], was designed and further implemented by the Thinking Machine Corporation, for processing semantic networks. The overall architecture consists of many (64K) processor/memory nodes interconnected as a hypercube. The concepts (nodes) of the semantic network are mapped onto the processor nodes of the architecture, and the interconnection between the processors represents the relationship between the corresponding concepts. All processors execute instructions from a single stream generated by a microcontroller under the direction of a conventional machine. Another parallel

architecture called the Semantic Network Array Processor (SNAP) is being studied at the University of Southern California [MOL85]. A square array of identical processing cells which are interconnected in the form of a mesh and also connected to a central controller constitute the SNAP architecture. The concept of mapping the data semantic network into an architectural interconnection is the same as in the Connection Machine. Complex searches and inferences are performed against the network by initiating the operations from many nodes simultaneously and by performing associative searches.

A semantic network machine called the IXM has been designed and simulated in Japan [FUR87]. IXM consists of an associative network with a large number of processing elements connected to it. Marker propagation, set operation, and

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association have been identified as basic operations in a semantic network. The processing elements include associative memories and are designed to execute the basic operations in parallel. The associative network consists of a number of network processors connected in a pyramid shape and the

network processors contain associative memories for supporting parallel marker propagation. The data network is partitioned into subnetworks and stored across the processing elements.

The user queries are issued in a semantic network language called the IXL, and the IXL commands can be interpreted by all the processing elements. The main disadvantage of these architectures is that their efficiency greatly reduces when

the semantic network cannot be directly mapped to the available main memory of the processing elements.

An asynchronous data flow model of computation has been proposed by Bic (B1C85) for processing semantic networks. The model is based on the idea of representing the semantic network as a dataflow graph in which each node is an active element capable of accepting, processing, and emitting data

tokens travelling asynchronously along the network arcs. Complex pattern matching is accomplished by representing the

query in the form of a message token and injecting it into the selected nodes of the graph. The token is propagated and matched across various nodes and links. Using this approach, no centralized control is required, and in addition to the parallel execution of a given request, multiple requests can

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be executed simultaneously. This increases the overall

performance of the system.

Semantic networks have been used for representing knowledge in domains with relatively smaller number of

objects. When the size of the semantic network grows, the data will have to be stored in the secondary storage. During processing, the data will have to be constantly staged in and out of the secondary storage and the efficiency of the

architectures mentioned above will reduce drastically. In our opinion, this problem can be tackled by increasing the granularity of the representation from the concept level to a class level. Many objects behaving similarly can be grouped

under the same class and the techniques developed can be applied at the class level.

The research presented in this dissertation deals with

the efficient processing of deductive queries against 00 knowledge bases with large sets of rules and complex data. The rules are structured and integrated into the 00 data. During processing, the structure facilitates in focusing on

the desired set of data and rules from the large knowledge base. In addition to the exploitation of OR parallelism, the various objects and/or relationships among the objects are derived in parallel and a distributed controlling mechanism is implemented.

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CHAPTER 3
REPRESENTATION AND QUERYING OF OBJECT-ORIENTED DATABASES


The limitations of record-oriented data models in capturing the complex structural relationships and the behavioral properties of objects in advanced application domains such as CAD/CAM have long been observed. Several Object-Oriented (00) semantic models have been developed to alleviate the limitations of record-oriented data models [HAM81, BAT85, HUL87, SU89]. The 00 semantic models provide a rich variety of modeling constructs, which simplifies the task of modeling complex data. The main features of an 00 data model are as follows:

(i) They support the unique identification of objects by system assigned object identifiers,

(ii) They allow the encapsulation of data and operations on the data,

(iii) They support abstract data typing and allow complex objects to be defined in form of aggregation hierarchies,

(iv) They allow the definition of generalization hierarchies (or lattices) and the inheritance of structural and behavioral properties among object classes in hierarchies.

In Section 3.1, we first present the 00 view of databases and illustrate the concept of a subdatabase, which is a 29

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structure for representing and processing 00 data. Further in Section 3.2, we illustrate the advantages of an 00 query model which is closed under the representation of subdatabases and present the operations and the philosophy of processing based on an 00 Query Language (OQL) which maintains the closure property [ALA89a, ALA89b]. Subsequently in Section 3.3, we illustrate with examples the various complexities of queries and their representation in OQL. We will also present with examples the features of a rule-based language (with OQL constructs) for processing deductive queries against 00 databases [ALA90]. The query processing algorithms presented in this dissertation are based on OQL.

3.1. Object-Oriented View of Databases


The 00 view of an application world is represented in the form of a network of object classes and associations (links) between these classes. We shall illustrate the concepts of 00 data representation using an example University database modelled by the 00 Semantic Association Model (OSAM*) [SU89]. Although OSAM* data model is used here, the data mapping and the query processing techniques presented in this dissertation are applicable to other 00 data models.

The University schema is shown in Figure 3.1. Using an 00 data model, objects within an application domain are uniquely identified by system-assigned Object Identifiers (OIDs) and objects with similar structures and behaviors are

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grouped together into classes. The rectangular boxes in

Figure 3.1 depict various classes of objects in the university domain. The interrelationships among these classes are represented by various types of associations which

characterize the retrieval and storage operation behaviors on their objects. Two of the widely recognized association types are shown in Figure 3.1., namely, Generalization (G) and Aggregation (A). An Aggregation association between two classes represents an attribute which can be visualized as a function that maps an object of one class to that of another. For example, as shown in Figure 3.1, the objects of a class Section are described by their section#s, textbooks, rooms,

Students, Teachers (of the. Section) and the Courses (to which they belong) The circles represent Domain classes from which the objects draw values of their descriptive attributes. The superclass-subclass relationship is specified by a

Generalization association between two classes. For example, in the f igure, Student and Teacher are subclasses of the class Person and inherit all the properties of the Person class. Similarly Grad and Undergrad are subclasses of Student and TA and RA are subclasses of Grad forming a Generalization hierarchy. It should be noted that the objects of a subclass

are a subset of the objects belonging to the superclass. Hence, an object plays different roles in the various classes

of the Generalization hierarchy. In order to distinguish the different roles of the same object we assign unique Instance

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Identifier (IID) to the individual instances of the object in the various classes. Objects in classes with no

Generalization relationships associated with it play a single role and each object has one instance identifier. In the

figure, similar associations are grouped together and labelled by A (for aggregation) and G (for generalization). The

various types of associations are treated consistently during the search process. They have different retrieval and storage operational behaviors. Objects of a class can be associated with objects of more than one class and a graphical view of an 00 database schema is represented by a network of interrelated object classes. A detailed description of the OSAM* model can be found in [SU89].

In the processing presented in this dissertation, the structure for the representation and processing of an 00 database is a subdatabase. A subdatabase is a part of the original database and is represented at the intensional and the extensional level as an intensional association pattern and a set of extensional association patterns respectively. Figure 3.2 shows an example subdatabase of the original database shown in Figure 3.1. The intensional association pattern of a subdatabase is represented by a network of Object classes and their associations as shown in Figure 3.2(a) which consists of classes Teacher, Section and Course and their associations. An extensional association pattern is a network of object instances and their associations that belong to

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the classes and association types of the intensional

association pattern. The set of extensional patterns of a subdatabase can be represented in the form of an extensional diagram. Figure 3.2(b) shows a possible extensional diagram of the example subdatabase. The t's, se's, and c's represent the unique Instance Identifiers (IIDs) of the objects of classes Teacher, Section and Course respectively. The

interconnection of t3 and se4 in the figure is an example of an extensional pattern, which records the fact that object instance t3 of class Teacher is associated with object instance se4 of class Section.

Each extensional pattern of a subdatabase can be classified as having one of the several extensional pattern types. An extensional pattern type is a common template that is shared by several extensional association patterns in a subdatabase. An extensional pattern type consists of a connected set of the subset of the object classes in the intensional pattern of the subdatabase. For example, the patterns of the subdatabase of Figure 3.2(b) belong to one of the following extensional pattern types shown in Figure 3.2(c). The extensional pattern type connecting classes Teacher, Section, and Course has as instances all the extensional patterns that connect the object instances of the classes Teacher, Section, and Course. The extensional

patterns that connect only the object instances of classes Teacher and Section belong to the extensional pattern type

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connecting classes Teacher and Section. Similarly, the

instances of the extensional pattern type connecting classes Section and Course can be explained.

3.2. A Closed Model of Ouery ProcessinQ
for Obiect-Oriented Databases


A "closed query model" can be defined as a model of query processing in which the structure of the output of a query is represented using the same data model with which the input of the query is structured. A closed model of query processing has several advantages. Since the result of a query is modeled by the same data model, it can be operated uniformly by another query using the operators of the same query language to further produce a new result. Also, the result of a query can also be saved as a view definition and manipulated uniformly as the original database. An Object-Oriented Query Language (OQL) [ALA89a, ALA89b], designed at the University of Florida, maintains the closure property for processing 00 databases. The query operates on one or more subdatabases and produces a new subdatabase.

The philosophy of the processing based on the OQL is to first identify the desired subdatabase and subsequently perform a set of specified operation(s) on the objects instances of the identified subdatabase. The search engine of the database management system establishes the desired subdatabase and then performs the operation(s). Thus a query block in OQL consists of two clauses, namely, a Context clause

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and an Operation clause. The Context clause has two optional subclauses: a Where subclause and a Select subclause. The structure is as shown below.

Context association pattern expression
Where conditions
Select object classes and/or attributes

Operation(s) object class(es)

The Context clause specifies the desired subdatabase by specifying the following in its association pattern expression: (i) the intensional pattern, (ii) the set of extensional pattern types, (iii) intraclass conditions, over the descriptive attributes of the object instances of various classes, qualifying the object instances. The interclass conditions are specified. in the Where subclause, and the desired descriptive attributes of the object instances of various object classes are specified in the Select subclause. A set of operations for the various classes of the subdatabase are specified in the Operations clause of the query. An

operation can be either a system-defined data manipulation operation (e.g., Display, Update, Print) or a user-defined operation (e.g., Rotate, Order-part, Hire employee).

The operators that can be used in the association pattern expression of the Context clause are the association operator and the nonassociation operator.

3.2.1. Association Operator

When the association operator (*) is applied to two directly associated classes A and Bin a database (i.e., the

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expression A B), it returns a subdatabase whose intensional pattern consists of the two classes A and B and their association. The resulting subdatabase also contains the set of extensional patterns drawn from the operand database such that each extensional pattern contains objects of both A and B. B objects that are not associated with any A objects and A objects that are not associated with any B objects in the operand database are not retained in the resulting subdatabase. The definition of the association operator can be easily generalized to the case when the association pattern expression contains more than two classes.

3.2.2. Nonassociation Operator

An exclamation sign (!) is used to denote this operator. When this operator is applied to two directly associated classes A and B in a schema (i.e., the expression "A I B"), it returns a subdatabase which contains only the instances of A that are not associated with any instances of B and the instances of B that are not associated with any instances of A.

3.3. Ouery Examples


The association operator has a higher precedence than the nonassociation operator. However, the precedence can be overridden by using parentheses. Various complexities of association patterns can be specified using the association and the nonassociation operators among the classes of the

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association pattern. We illustrate the various complexities of the association patterns with example queries. The queries are described for the schema shown in Figure 3.1. The english language description and the OQL representation are specified for each example query.

3.3.1. Noncyclic Association Pattern

In this section we consider queries whose association patterns do not form cycles. We classify noncyclic association patterns into two types namely, linear association patterns and branching association patterns.

3.3.1.1. Linear association pattern

This is the simplest form of the structure of the association pattern. The various classes specified in the association pattern are related in a linear string. The

following query 1, query 2, and query 3 are example of queries with a linear intensional pattern. Query 1: For all the Courses with courses greater that C600, and being offered by the Departments in the college of Engineering, and having currently offered Sections with section#s either less than S250 or greater than S550, retrieve the course#s of the Courses, the name of the Department offering the Courses, and the sections of the Sections.



Context Department [college = 'Engineering'] *
Course [course# >6000] *
Section[section# < S250] OR [section# > S550] Retrieve Course (course#], name, section#

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The association pattern of the desired subdatabase in this query is a linear string of object classes Department, Course, and Section. The classes of the association pattern are associated with the association operator (*). The query specifies a retrieval operation on the resulting subdatabase. The descriptive data values of the course#, name, and section# attributes are to be aggregated under the selected objects of class Course.

Query 2: For all the currently offered Sections with enrolled Students who have not decided on a majoring Department, obtain the names of the Departments, and also the section#'s of the Sections, and the classification of the enrolled Students. Context Section Student Department Retrieve Department (name] ;
Section sections#s, classification

The association pattern of the desired subdatabase is a linear pattern of object classes Section, Student, and Department. The classes are related using both the association and the nonassociation operator. It should be noted that the association operator has precedence over the nonassociation operator. Two separate relations are to be retrieved. The first relation contains the names of all the Department objects in the resulting subdatabase. The second relation contains the sections#'s of the selected Section objects and the classification of the selected Students aggregated under the objects of class Section.

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Query 3: For all the Students with no majoring Department, and enrolled in currently offered Sections obtain the section's of the Sections and the classification of the Students. Also obtain the names of the Department with no majoring Students.

Context Section (Student Department) Retrieve Department [name], Section sectionsf], classification

The association pattern of the resulting subdatabase consists of classes Section, Student, and Department as in query 2 above. Also, the structure and the operators among the object classes are the same as in query 2. However, the precedence of the association operation over the nonassociation operation has been overridden by the use of parenthesis. The structure of the result desired is also similar to that of query 2.

3.3.1.2. Branching association pattern

An association pattern expression may contain branches expressed by an AND or an OR operator. The following query 4, and query 5 are example of queries with a branching

intensional pattern.

Query 4: For all the currently offered Sections taught by a Teacher with a Ph.D. degree, and enrolled by Students who are Graduate Students, and of Courses being offered by the 'CIS' Department, retrieve the section's of all the Sections and the degree of all the related Teachers and the course's of all the related Courses.

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Context Teacher [degree = 'Ph.D.') Section AND
(Course Department [name = 'CIS']
Student Grad)

Retrieve Section [section#), degree, course# The association pattern of the desired subdatabase in this query is a branching pattern of object classes Teacher, Section, Course, Department, Student, and Grad. The object class Section, at which the branching occurs is called the fork class. An AND operator is specified between the branches of the fork class. An AND operator means that in the result, an instance from the fork class must be associated with instances from all the classes related with the forking branches. The section's of all the selected Sections and the degree of the related Teachers, and the course's of the related Courses are to be retrieved from the resulting subdatabase.

query 5: For all the currently offered Sections taught by a Teacher with a Ph.D. degree, and either enrolled by Students who are Graduate Students, or of Courses being offered by the 'CIS' Department, obtain the section's of all the Sections, the degree of the related Teachers, and the course #'s of the related Courses.

Context Teacher (degree = 'Ph.D.'] Section OR
(Course Department [name = 'CIS'],
Student Grad)

Retrieve Section [section#], degree, course# The association pattern of the desired subdatabase in this query is also a branching pattern of classes Teacher, Section,

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Course, Department, Student, and Grad. Also, similar to query 4, the object class Section is the fork class. However, an OR operator is specified between the branches of the fork class. An OR operator means that, in the result, an instance from the fork class must be associated with an instance from at least one of the two related branching classes. The structure of the desired result is the same as in query 4.

3.3.2. Cyclic Association Pattern

The association pattern can also contain cycles. The

following is an example of a query with a cyclic association pattern.

Query 6: For all the Sections being taught by a Teacher with a 'Ph.D.' degree, and belonging to Courses being offered by

the Department which has Students (who are currently enrolled in those Sections) major in, retrieve the textbook taught by the Section and the course# of the related Courses. Context Teacher [degree = 'Ph.D. ') Sec tion AND (Course Departmuent, Student) AND* Grad
Retrieve Section [textbook], course#!

The association pattern of the desired subdatabase in this query consists of object classes Teacher, Section, Course, Student, and Department. The object classes are associated with the association operator. Moreover, branching occurs at

object classes Section and Department which are the fork classes. An AND operator is specified between the branches of both the forking classes and a cyclic association pattern in

formed. The textbook of the selected Sections and the course#

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42

of the related courses are to be aggregated under the Section objects in the result.

3.3.3. Deductive Oueries

New subdatabases can be derived from other existing or derived subdatabases. A derived subdatabase is called the target subdatabase and the subdatabases used to derive it are called the source subdatabases. The process of derivation is captured by the derivation rules. A derivation rule has an IF-THEN structure as follows:

IF Context association pattern expression
Where conditions

THEN subdatabase-id (association pattern expression) The Context clause and the its optional Where subclause are the same as described in Section 3.2. above. The subdatabaseid in the THEN clause is a unique name to be given to the derived subdatabase. The intensional pattern of the derived subdatabase consists of a subset of the classes referenced in the association pattern expression of the IF clause. Other unreferenced classes will not be retained in the derived subdatabase. The extensional patterns of the new subdatabase are derived from the extensional patterns that satisfy the conditions of the IF clause and its Where subclause. The

following is an example of a deductive rule. Rule 1: New relationships establishing the fact that good quality Teachers are teaching good Students taking high level Courses can be established, if the Teachers who have a Ph.D. degree are teaching the Sections, in which the Students with

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GPA higher than 3.5 are enrolled, and these sections belong to Courses having course# greater than 6000.


IF Context Teacher [degree = 'Ph.D.'] Section *
AND (Student [GPA > 3.5],
Course [course # > 6000])

THEN good (Teacher Student Course)


This rule when executed against the database of Figure 3.1, returns a subdatabase whose set of extensional patterns are of the type . It should be noted that the relationships in the new subdatabase are derived and are not present in the original database. Also, the objects of class Section are not retained in the new subdatabase because the object class Section is not referenced in the association pattern expression of the THEN clause.

Once the deductive rule(s) that derive a new subdatabase are defined, the classes of the derived subdatabase can be referenced in association pattern expressions in any OQL query in the normal way. For example, the following query

references the classes in the subdatabase defined in the THEN clause of the rule above.

Query 9: For all the good Students majoring in the college of Engineering and enrolled in high level courses, retrieve the title of the Courses, the GPA of the enrolled Students, and the name of their majoring Department. Context Department [college = 'Engineering'] good:Student
good:Course


Course [title], GPA, name


Retrieve

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The association pattern specified in this query references an association (between Student and course) which is not

explicitly stored in the original database. However, the association can be derived from the original database by executing the derivation rule specified above. The execution of this query would trigger the execution of the rule. Once the rule derives the desired association, the query can be executed to establish the database specified in the Context clause of the query.

The execution of the rule may itself trigger other rules for deriving source database(s) of the rule and an inference chain will be established. Also, more than one rule can derive the extensional patterns of the same subdatabase. When more than one rule for the same subdatabase is specified, all the rules are executed and a union of the extensional patterns derived by the individual rules is considered for further processing of the derived subdatabase.

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0
SS#\Al Person[ name


A-A


Teaching Research Assistant Assistant
(TA) (RA)


name college


Figure 3.1 An University Schema

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eac her Section o Course




(a) The Intensional Pattern of a Subdatabase


(b) An Extensional Diagram of the Subdatabase Teacher Section Course


Teacher Section


Section Course




(c) Extensional Pattern Types of the Subdatabase Figure 3.2 The Specification of a Subdatabase

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CHAPTER 4
PARALLEL ARCHITECTURAL MODEL AND DATA ORGANIZATION


In this chapter, we discuss the desired features of a parallel architecture for the efficient implementation of large object-oriented databases. Further, we illustrate

techniques for partitioning the large sets of complex data and organizing them across the nodes of the parallel architecture. The main objective behind the data partitioning techniques is to reduce the overall query execution time. In Section 4.1 we present the parallel architectural model and in Section 4.2 we discuss the data organization. The data partitioning and mapping techniques presented in this chapter have been experimentally analyzed and the results are presented in Chapter 7.

4.1. Parallel Architectural Model


Querying on large and complex Object-Oriented databases

involve retrieving and manipulating data about various object classes. The number of object instances in each class, the

amount of data about each object instance, and the associativity among the individual object instances can be

enormous in large databases. The data has to be stored across several secondary storage devices. Moreover, data about the 47

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selected classes of object instances have to be interrelated based on the explicitly captured associations. During

processing, large quantities of data have to be retrieved from several secondary storage devices and transferred among the processing nodes of the system.

Shared memory architectures are not suited for this type of processing since, at high data rates, memory contention drastically reduces performance. Message passing systems are a promising alternative provided (a) the processing nodes have sufficient processing power and storage capability to store and process the large sets of data, and (b) the bandwidth of the interconnection network is suitable to handle the communication among the processing nodes.

Figure 4.1. shows the model of a parallel system considered in our study. It consists of a set of processing nodes, each containing a processing unit, main memory elements, and several secondary storage devices. The

processing nodes are interconnected by a regularly and homogeneously connected interconnection network. Since data retrieval is one of the dominant factors in database processing, parallel 1/0 at each node improves the retrieval

parallelism. In a regularly connected system, each processing node is directly connected to the same number of other

processing nodes. A homogeneous system has topologically identical processing nodes and the connection structure at each node repeats in a regular fashion. The topological

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similarity and the regularity among the interconnection

components at each processing nodes reduces the development costs particularly for a significant number of units. Also, the configuration can be easily expanded when the processing demand increases.

The maximum delay among any two processing nodes in the system varies with the exact topology of the system. The overall bandwidth of the network varies with the degree of connectivity of the nodes in the system. The data

partitioning and mapping algorithms presented in the next section and the query processing algorithms presented in Chapter 5 and Chapter 6 are not dependant on the topology and

can be executed with varying performances on different system topologies.

The database is partitioned and stored across the various secondary storage devices of the processing nodes in the system. As can be seen in Figure 4.1, the user is interfaced

by one or more host processors, which are connected to the processing system. The user issues queries at one of the host processor. The query is compiled into a set of messages and transferred to the relevant processing nodes in the system.

The processing nodes retrieve and manipulate the pertinent data from their secondary storage devices. In addition, the

processing nodes pass data among each other during the course of query processing. Finally, the result is transferred to the host processor for presentation to the user.

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4.2. Partitioning~ and Maprinaf of Data



The physical organization of the data across the

processing nodes of the system plays an important role in determining the overall execution time of a query. A data organization scheme can improve the query execution time in a variety of ways. Firstly, in a multicomputer system, the data can be accessed by a processing unit faster from its local secondary storage devices than from remote devices. Hence, reduction in data retrieval time can be accomplished by

placing similarly accessed data together across the secondary storage devices of either a single processing node or across a set of closely connected processing nodes. Secondly, by organizing the different data segments in such a manner so as to balance the processing load among the cooperating processing nodes, a reduction in the query execution time can be accomplished. Moreover, resource utilization can also be

improved by load balancing. Thirdly, when multiple processors are used cooperatively to answer a query, data communication

among the processors can potentially account for a significant portion of the query execution time. By intelligently mapping the data segments across the processing nodes, the average number of hops taken by the data while travelling from the sending processor to the receiving processor can be reduced. This in turn reduces the overall communication costs during query processing.

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In this section, we present a methodology for organizing the complex data of large Object-oriented databases across the processing nodes of the parallel model architecture presented in section 4.1. The methodology is presented with an

illustrative example. As a first step, based on the knowledge of the database schema, data clusters are formed and the pattern of communication among the data clusters is determined. A data cluster consists of the descriptive and

the associative data about either all or a subset of the object instances of an individual object class. In the

initial clustering phase, a data cluster consists of data about all the object instances of an object class. The

computation cost associated with each data cluster, and the

cost of data communication from each data cluster to other associated data clusters is estimated based on the data characteristics.

Subsequently, depending on the total number of processing nodes in the system and the total amount of data in various data clusters, the data are organized to form groups of data clusters. The number of groups equal to the number of

processing nodes in the system, and the groups are formed such that the computation load associated with each group is nearly the same. During this load balancing phase, a data cluster with large amount of data is partitioned to create new data clusters each with relatively small amount of data. In

addition to load balancing, the grouping is performed so as to

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allow the parallel processing of the queries with minimal increase in communication overheads. Finally, the groups of data clusters are mapped or assigned to the processing nodes of the system. The groups are mapped such that the

communication costs among the processing nodes is reduced during query processing.

4.2.1. Data Clustering

During query processing, the desired data about all the

object instances of a referenced object class are retrieved and processed similarly. Thus, all the data pertaining to an object class can be clustered and stored together in order to improve localization. Thus we def ine a data cluster as containing all the descriptive and the associative data about either a subset or all the object instances of a single object class. Figure 4.2(a) and Figure 4.3(a) show example data clusters, in the form of nonnormalized relations, pertaining

to the object classes Section and Teacher of the schema, shown in Figure 3.1 respectively. The network data of the database

can be partitioned as multiple nonnormalized relations for the individual classes of the database. It should be noted that

nonassociation of an object with object(s) from other classes is not stored as null values. The relationship itself is not stored. During query processing the relationships that are

present are used in computing the desired subdatabase. As can be seen from Figure 3.1, the object instances of class Section are described by their section#, textbook, and room#, and are

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associated with the object instances of object classes Teacher, Student, and Course. The Section lID in the first

column of the relation in Figure 4.2(a) represents the instance identifiers of the object instances of the Section

class. The second, third, and fourth columns of Figure 4.2(a) represent the values of the section#t, textbook, and the room# attributes respectively of the object instances of the object class Section. The relationships among the object instances

of class Section and the object instances of classes Teacher, Student, and Course are captured and explicitly represented in the fifth, sixth, and seventh columns respectively of Figure 4.2(a). Similarly the population of values in the data cluster of Figure 4.3(a) can be explained. Moreover, any

specified operations on the object instances of an object class are stored along with the declarative data of the object class.

Further, in order to improve the retrieval parallelism,

we vertically partition the nonnormalized relations of the individual object classes into binary relations. Figure

4. 2(b) and Figure 4.3 (b) represents the vertical partitions of the nonnormalized relation of Figure 4.2(a) and Figure 4.3(a) respectively. During query processing, values of a certain specified subset of the attributes of the object instances of a class are desired. Also, the associations among the object

instances of a class with the object instances of a subset of the related classes are manipulated during the course of query

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processing. By vertically partitioning the data and storing

them separately, specific partitions can be retrieved and the retrieval of unnecessary data can be avoided. Also, different vertical partitions can be retrieved in parallel thereby

improving the retrieval parallelism. The scheme of vertically partitioning the data is similar to the one proposed for relational systems (VAL87].

The data clustering and vertical partitioning scheme proposed above improves query execution time by localizing retrievals and reducing the amount of unnecessary data retrieved. However, the total amount of data stored is increased. The relationship data between the object instances of two associated classes is replicated in the nonnormalized

relations of both the related classes. For example, as can be observed from Figure 4.2(a) and Figure 4.3(a), the relationships between the object instances of classes Teacher and Section are replicated in the data clusters of both the classes. Also, as can be observed from Figure 4.2(b) and Figure 4.3(b), the instance identifiers of the object

instances of a class are replicated in all the vertical partitions of the class.

The clustering scheme creates a data cluster for each object class. In a steady state, under the assumption that

queries involving various object classes and relationships among object classes have equal probability of occurrence, the data retrieval and processing time associated with a data

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cluster is proportional to the amount of data in that cluster. Also, as will be evident from the description of the query processing algorithms in Chapter 5 and Chapter 6, data from a cluster is related with the data in other related data cluster(s). The amount of data communicated from a sending

cluster to the receiving cluster is proportional to the number of object instances in the sending cluster and the average number of object instances of the receiving cluster which are associated with each object instance of the sending cluster. We represent the computation costs of a data cluster in terms of the total number of bytes of data in it. The cost of

communication from a sending data cluster to a receiving data cluster is represented in terms of the number of object instances transferred from the sending data cluster to the receiving data cluster. The steady state computation and communication costs are represented as a computationcommunication graph. As stated above, the costs of computation and communication are computed in this

dissertation for an identical frequency of queries referencing the various parts of the database. Nevertheless, the same methodology can be used for computing costs when the frequency of queries referencing different segments of the database varies and is known. A computation-communication graph is a directed and weighted graph. Each vertex of the graph

represents either a single data cluster or a group of data clusters. A directed edge from an originating vertex to the

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directed vertex depicts the direction of data communication

f rom the data cluster (or group of data clusters) represented by the originating vertex to the data cluster (or group of data clusters) represented by the directed vertex. The weight of a vertex represents the total computation cost associated

with it and the weight of the directed edge represents the communication cost of sending data from the originating vertex to the directed vertex.

We illustrate the process of determining the computationcommunication graph with an example database. Figure 4.4 shows the schema of the example database. The values of various parameters characterizing the database are shown in Table 4.1 and Table 4.2. The same example database will be used to illustrate the subsequent phases of the data organization methodology. For simplicity, the size of the values of the descriptive attributes in the example database

is assumed to be 10 bytes. Also, the size of the total amount of stored data for each object instance of all the object classes is assumed to be 200 bytes. The derived computationcommunication graph for the example database is shown in Figure 4.5. The vertices of the graph represent the computation associated with various data clusters in the database. It should be noted that at the end of the initial clustering phase the data about individual object classes is

clustered together and the number of vertices of the graph equal the number of object classes in the database. The data

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clusters are represented as Cis. The number within each vertex represents the total size of the data within each cluster in Mega bytes. The directed edges of the graph represent the direction of communication of data among the related clusters. The number along side each edge represents the total number of object instances, in thousands, communicated along the edge.

The following formulae are used in computing the computation and communication costs:

Let the number of object instances in the object class c be NObjcts(c).

Let the number of descriptive attributes of object class c be N-Desc-Attrs(c).

Let the set of classes associated with the object class c be Assoc-Classes-Set(c).

Let the size of the value of the descriptive attribute a of an object class c in bytes be Size-Desc-Attr(c,a). Let the average number of object instances of object class c associated with each object instance of class j be AvgConn(c,j).

Let the size of the instance identifier in bytes be Size-Id. The size of the total amount of data stored for each object instance of an object class c is

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Data-Per-Objct(c) = Z [i= 1 TO N-Desc-Attrs(c)] (Size-Id + Size-Desc-Attr(c,i)) +

Z[ V j c Assoc-Classes-Set(c)J (Size-Id + Size-Id Avg-Conn(c,j)).

As was mentioned earlier, the data about an object instance are vertically partitioned, and each partition contains the instance identifier and the attribute value or an instance identifier and the instance identifiers of the related class. Thus, the size of the total amount of data in a cluster containing the data about all the object instances of an object class c is

Data-Per-Class(c) = Data-Per-Objct(c) N-Objcts(c) Data-Per-Class(c) is also the computation cost associated with the data cluster pertaining to object class c. The total number of object instances transferred from the data cluster pertaining to object c to the associated data cluster pertaining to object class j is N-IID-Xfer(c,j) = N-Objcts(c) Avg-Conn(c,j) N-IID-Xfer(c, j) is also the communication cost associated with the directed edge originating from the cluster pertaining to object class c, and pointing to the cluster pertaining to object class j. Since a data cluster contains data about a single object class, we will interchangeably use the above definitions for an object class and a data cluster pertaining to the object class.

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4.2.2. Load Balancing

By storing all the data within a data cluster across the SSD(s) of a single processing node localization can be improved. Also, by storing the various data clusters across different processing nodes the data in individual data

clusters can be accessed in parallel during query processing. However, the number of object instances and the size of the

data about each object instance varies with the object class, thereby, varying the amount of data in each cluster. Due to the varying amount of data in different clusters, the different cooperating processors will take varying amount of

data retrieval and processing time. Also, the number of processing nodes in the system can be different from the number of object classes in the database. Query execution time can be reduced by balancing the data retrieval and processing across the processing nodes of the system.

One possible method of load balancing is to horizontally partition each cluster equally among the available processing nodes of the system. However, this balancing scheme restricts the amount of processing parallelism and increases the communication costs during query processing. Using the above

partitioning scheme, different horizontal data segments about an object class will be processed in parallel by all the processing nodes of the system. However, the data has to be sequentially related from one object class to another and the desired subdatabase has to be established in repeated cycles

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of forward and backward propagation depending on the complexity of the query. As will be evident from the

description of the parallel query processing algorithms in Chapter 5 and Chapter 6, a query can be processed in parallel by processing data simultaneously from various object classes referenced in the query. The parallel processing algorithms eliminate the complexities involved in sequentially relating

data from one object, class to another. Also, using the above partitioning scheme, the data about all the object classes is distributed among the processing nodes. At every processing step, data from each processing node has to be replicated and

transferred to all other processing nodes. This in turn increases the overall communication costs and consequentially the query processing time.

We balance the data clusters among the processing nodes of the system by horizontally partitioning the data of those clusters having large amounts of data, and by grouping together clusters having relatively small amount of data. The optimal amount of data per processing node for balanced data retrieval is estimated and the reorganization is performed in two steps. During the first step, data clusters with data mo re than the optimal value are partitioned into new clusters with data less than or equal to the optimal value. During the second step, original and new clusters having data

less than the optimal value are grouped to create several groups of data clusters. The groups are created such that the

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combined data in each group is closer to the optimal value. At the end of the load balancing phase, the number of groups equal the number of processing nodes in the system. The data is reorganized so as to allow the parallel processing of the query with minimal increase in communication overheads.

The following formulae are used in determining the optimal amount of data in each group of data cluster(s). Let the number of object classes in the database be N-Classes. Thus, the total amount of data stored in all the clusters of the database is

Total-Data = [ [i=l TO N-Classes] (Data-Per-Class(c))

Let the number of processors in the system be N-Prcs. The desired size of the data per processor after load balancing is

Data-Per-Proc = Round (Total-Data / N-Prcs).

The Data-Per-Proc is the optimal amount of data desired in each group of cluster(s). The computed values of the total data size and the desired data per processor for the example database of Figure 4.4 is shown in Table 4.2.

4.2.2.1. Phase I Rartitioning of clusters

During this phase, the clusters having data more than the desired data per processor are horizontally partitioned. The following presents the formulae and the algorithmic step of the partitioning phase. The partitioning of the data clusters of the example database is also illustrated.

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The number of object instances of class c, such that the data about that number of object instances equals the desired data per processor is

Optimum-N-Objcts(c) = Truncate (Data-Per-Proc / Data-Per-Objct(c)).
Let the number of data clusters at any instant of the partitioning phase be N-Clusters. It should be noted that at the beginning of the partitioning phase the number of clusters equal the number of classes. Let each individual cluster be denoted by an unique integer from

1 to N-Clusters.

For i = 1 To N-Classes

If (N-Objcts(i) > Optimum-N-Objcts(i)), then

partition the data cluster pertaining to class i into N-part(i) clusters. The partitioning is

performed such that (N-Part(i) 1) clusters contain all the data about Optimum-N-Objcts(i) object instances of the class i and the last cluster contains all the data about (N-Objcts(i) (N-Part(i) 1) Optimum-N-Objcts(i)) instances.

N-part(i) = Truncate (N-Objcts(i) /

Optimum-N-Objcts(i))

The total number of data clusters is increased

appropriately.

N-Clusters = N-Clusters + (N-Part(i) 1)

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Table 4.3 shows the cluster names and total data in each cluster at the end of the partitioning phase. A cluster name

of C (i, j) in the table refers to the cluster belonging to the jth partition of class C(i). The optimal data per cluster is 2.2 M bytes. It should be observed that the cluster C(1) had

more than optimal data and was split into two clusters, namely into C(1,1) and C(1,2).

4.2.2.2. Phase II grouping of clusters

During this phase, all the clusters, among the clusters

at the end of the partitioning phase, whose data are less than the desired data per processor are organized to create groups of clusters each with the desired amount of data. Also, the clusters which have the desired data per processor are organized as groups of one cluster each. It should be noted that at the end of the grouping phase, the number of groups

equal the number of processing nodes. At each step of the grouping process, an estimation is made about the communication costs that would be incurred during processing when two potential clusters are grouped. Among the possible grouping choices, the clusters that incur minimal

communication costs are grouped. The following illustrates the grouping process. The grouping of the data clusters of Table 4.3 is also shown as an illustrative example.

Let the set of cluster groups that contain the desired size of the data at any instant be Optimum-Group-Set.

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After the grouping of clusters, a cluster group may contain clusters from more than one object class. Let the number of object instances of an object class c in a cluster group G(i) be N-Objcts(c,G(i)). Let the set of object classes to which the data in a cluster group G(i) belongs be Class-Set(G(i)). Let the function returning the class of a cluster C be Class(C).

Step 1: Identify all those data clusters, resulting from the partitioning phase, that contain the data about the optimal number of object instances of the relevant class. Assign the identified clusters to the Optimal Group Set. The following pseudo code illustrates the step. i=0

Optimal-Group-set = { )

For C = 1 To N-Clusters

If (N-Objcts(C) = optimum-N-Objcts(Class(C)), then

i=i+l G(i) = (C)

Optimum-Cluster-Set = Optimum-Cluster-Set + G(i)

The Optimum-Cluster-Set identified from the clusters of Table

4.3 is

Optimum-Cluster-Set =( G(1) = (C1,1), G(2) = (C1,2), G(3) = {C4), G(4) = (C5), G(5) = (C7), G(6) = (Ci) ).

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Step 2: For all the clusters not in the Optimum Group Set, assess the communication cost associated with the cluster and identify the cluster with the minimal communication cost.

During processing, data from a cluster of an object class are related to the data from a cluster of another object class that is associated in the schema. As can be observed from the description of the algorithms in Chapter 5 and Chapter 6, the intensity of communication for relating object instances from a cluster of an originating class to a cluster of the related class is proportional to the number of the object instances in the originating cluster belonging to the originating class and the average number of object instances of the related class associated with each instance of the originating class. The amount of data transmitted from a data cluster C to another associated data cluster J is proportional to Data-Trans(C,J) = N-Objcts(C) Avg-Conn(Class(C), CLass(J)) It should be noted that Data-Trans(C,J) is not equal to DataTrans(J, C). Also, in order to increase the flow of data, the processor storing the clusters are bidirectionally connected. Thus, the communication cost due to the transmission of data among two clusters C and J is

Comm-Cost(C,J) = Maximum(Data-Trans(C,J), Data-Trans(J,C)) The total communication cost associated with the data cluster C due to the transmission of data among all the clusters related to C is

Comm-Cost(C) = Z [Y I c Assoc-Classes-Set(C)] Comm-Cost(C,I)

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The above mentioned formulae are used in computing the communication costs of all the relevant clusters.

Subsequently, the cluster with the minimum communication cost is identified. Table 4.4 shows the communication cost associated with the clusters of Table 4.3 which could not be

assigned to the Optimal Group Set. As can be observed,

cluster C10 has the lowest communication cost. Step 3: Estimate the cost of grouping the cluster with the minimum communication cost (obtained from step 2), say C-Mmn, with each of the clusters not assigned to the optimum cluster set, and determine the cluster with minimum cost of grouping.

Two cases arise depending on the combined data size after grouping the two clusters. When the combined data size is

less than the desired data per processor after balancing, the clusters can be grouped in their entirety. However, when the combined data size is more than the desired data per processor after balancing, the cluster being grouped is partitioned into two new clusters, and C-Mmn is grouped with one of the new clusters. The grouping cluster is partitioned such that the combined data of the C-Mmn and one of the new clusters equals the desired data per processor. The following pseudo code illustrates-the step.

Let the total size of the data in a cluster group G(i) be Data (G(i)).

The size of the data in the cluster group G(i) is

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Data(G(i)) = Z [ c e Class-set(G(i))] (N-Objcts(c,G(i))

Data-Per-Objct(c)

The amount of data transmitted from the cluster group containing the two clusters I and J to the cluster K is proportional to

Data-Trans((I,J),K) = Data-Trans(I,K) + Data-Trans(J,K).

Similarly, the amount of data transmitted from the cluster K to the cluster group having the clusters I and J is proportional to

Data-Trans(K,{I,J)) = Data-Trans(K,I) + Data-Trans(K,J).

Let the cost of grouping two clusters I and J be GroupCost(IJ).

For every cluster (except the one with the minimum communication cost), say J, which is not assigned to the Optimum Group Set, the following steps are executed and the cluster with the minimum grouping cost is obtained. Case 1: If (Data(C-Min + J) 5 Data-Per-Proc), then Grouping-Cost(C-Min, J) = [ (C = 1 TO N-Clusters) & (C o C-Min) & (C o J)]

Maximum (Data-Trans({C-Min,Cj),Ck) Data-Trans(Ck,(C-Min,cj))

Case 2: If ( Data(C-Min +J) > Data-Per-Proc), then The data cluster J is partitioned into two clusters, say Jl and J2, such that

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N-Objcts(Jl) = Truncate ((Data-per-Proc Data(C-Min) / Data-Per-Obj ct (J))

and N-Objcts(J2) = (N-Objcts(J) N-Objcts(Jl)).

The increased communication cost due to the partitioning of the cluster J into clusters JI and J2 is Split-Cost(J) = (Comm-Cost(Jl) + Comm-Cost(J2) Comm-Cost(J))

Hence, the overall grouping cost in this step are the sum of the cost of grouping the cluster C-Min with J1, and the cost of partitioning the cluster J into clusters J1 and J2. Grouping-Cost(C-Min,Jl) = Z [ (K = 1 TO N-Clusters) & (K C-Min) & (K 0 J1)

Maximum (Data-Trans((C-Min,Jl),Ck) ,

Data-Trans(Ck,{C-Min,Jl)) + Split-Cost(J)

Step 4: Group the two clusters C-Min and the cluster with the minimum cost of grouping (obtained from step 3), say C-Merge, and if the combined data in the new group equals the desired data per processor then add the new group to the Optimum Group Set. Repeat the process of determining the cluster, among the clusters not in the Optimum Group Set, with minimum communication cost and grouping it with other clusters. However, if the combined data in the new group is less than the optimal data then repeat the process of adding other clusters to the new group. Terminate the process of grouping when the number of groups in the optimal group set equal the

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number of processors. The following pseudo code illustrates the step.

Let the new merged cluster group be G(new). If C-Merge is one of the original clusters and is not created due to the partitioning of an existing cluster in Step 3 above then N-clusters = N-Clusters 1 If (Data(C-Min,C-Merge) = Data-Per-Proc), then

Optimum-Group-Set = Optimum-Group-set + G(new) If (N-Clusters # N-Prcs), then

If (Data(C-Min,C-Merge) = Data-Per-Proc), then

Go to Step 2

If (Data(C-Min,C-Merge) < Data-Per-Proc), then

Go to Step 3

Let us consider the grouping of the clusters, which are not in the Optimal-Group-Set, of Table 4.3. Table 4.4 shows the relevant information about the clusters with less than the optimal data per processor. The name of the cluster, the total data in each cluster, the names of the clusters associated with each cluster, and the communication cost associated with each cluster are shown in Table 4.4. The communication costs is represented in number of object instances communicated among the clusters and are computed based on the formulae shown in step 2 above. As can be seen from the table, cluster C10 has minimal communication cost associated with it and hence is grouped first. Table 4.5 shows the cost of grouping C10 with other clusters having less

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than optimal data. Table 4.5 also shows the cost of splitting clusters wherever appropriate. For example, clusters C2 and C3 cannot be grouped in their entirety and have to be partitioned. The grouping cost includes the cost of partitioning. The costs are estimated based on the formulae shown in Step 3 above. As can be seen from Table 4.5, among the possible grouping choices, the grouping of a partition of the cluster C2 with the cluster C10 adds the least communication cost. Cluster C2 is partitioned into two clusters C2,1 and C2,2. Cluster C2,1 contains data about 5,000 object instances of the object class 0C2, and the cluster C2,2 contains data about 4,000 object instances of the object class OC2. The cluster C10 is grouped with the clusterC2,1. The combined data of the two clusters equals the optimal data desired per cluster and the new cluster is assigned to the Optimal-Group-Set. The new Optimal-Group-Set is

Optimal-Cluster-Set = (G(l) = {Cl,l), G(2) = (Cl,2), G(3) =

(C4), G(4) = (C5), G(5) = {C7), G(6) = (Cll), G(7) = (Cl0, C2,1) ).

Since the number of clusters in the optimal group set does not equal the number of processors, the grouping process is continued. Table 4.6 shows the names of the clusters which are not in the Optimal-Group-Set. The total size of each cluster, the set of associated clusters, and the communication cost associated with each cluster is also shown in the table.

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The communication costs are computed based on the formulae shown in step 2 above. As can be seen from Table 4.6, the cluster C2,2 has minimum communication cost associated with it and is grouped next. Table 4.7 shows the cost of grouping the cluster C2,2 with other clusters with less than optimal data. Since no cluster involves partitioning during the grouping process, the splitting cost is zero for all the clusters. The grouping costs are estimated based on the formulae shown in

step 3 above. As can be seen from the Table 4.7, the grouping of the cluster C2,2 with the cluster C6 adds the least

communication cost and hence the clusters are grouped. The

combined data of the cluster group containing the two clusters does not equal the optimal data per group and hence the new group is not assigned to the optimal group set and is grouped further.

Table 4.8 shows the cost of grouping the cluster

group(C2,2, C6) with the remaining clusters not in the Optimal -Group-Set. The split cost for-clusters which require partitioning of the cluster before grouping is also shown in

the table. The costs are computed based on the formulae shown in step 3 above. As can be seen from Table 4.8, the grouping

of cluster C8 with the cluster group {C2,2, C6) adds the least communication cost and hence the clusters are grouped. Also,

the combined data of the new group of clusters equals the optimal data desired per processor. Similarly, the remaining two clusters, C8 and C9 are combined to create a new group of

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clusters with optimal data in it. The final Optimal-ClusterSet is, Optimal-Cluster-Set = { G(1) = (Cl,l), G(2) = {Cl,2), G(3) = {C4), G(4) = (C5), G(5) = (C7), G(6) = (C11), G(7) = (CIO, C2,1), G(8) = {C2,2, C6, C8), G(9) = {C8, C9) ). Figure 4.6 shows the final computation-communication graph. It

should be noted that each node of the graph contains the same amount of data and has the same retrieval and processing time associated with it. The clusters forming each cluster group of the computation-communication graph, and the total Mega bytes of data in each of the clusters is shown inside the circles representing the cluster groups. Unique vertex number is also assigned to each cluster group. The number alongside the edges of the graph represent the total number of object instances, in thousands, communicated among the communicating cluster groups.

4.2.3. Mapping of Cluster Groups Onto Processors

The load balancing phase creates data cluster groups with nearly equal amount of data. Also, the number of data

clusters groups equals the number of processing nodes in the system. By mapping one cluster group per processor, the data can be evenly distributed in the system. During processing, data from one cluster group is related to data from other cluster group(s) that contain the data pertaining to the associated class(es) of the class set of the original cluster group. The pattern and the intensity of data communication among the cluster groups is irregular in nature. The

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computation-communication structure resembles a weighted irregular directed graph. The nodes in the graph will represent the time for the retrieval and manipulation of the data of the individual cluster groups. Since the data among

the cluster groups was balanced in the previous phase, all the nodes of the graph will have the same time associated with them. A directed arc from an originating node to the directed node in the graph will represent the communication of data from the originating cluster group to the directed cluster group. The weight associated with the arc will represent the

amount of data transmitted from the originating cluster group to the directed cluster group.

An optimal data placement of irregularly communicating

data cluster groups across the processing nodes of a parallel system requires the processing nodes to be fully connected. However, due to cost and other technical considerations

processing nodes cannot be fully connected and are usually connected in a regular fashion. Mapping of cluster groups with an irregular communication patterns among them onto a set of regularly connected processing nodes with the objective of optimally minimizing the overall communication costs is

similar to the optimal mapping of the irregular computationcommunication graph onto a regular graph of processing nodes. The latter mapping has been shown in the literature to be NP complete (LOBS, GAR79].

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It is necessary to develop appropriate application specific heuristic methods to obtain suboptimal mapping. Researchers in the past have taken different approaches in obtaining mapping of problem graphs on parallel architectures for various applications [BOK8l, SAD87, BOK88]. Recently, Baru [BAR9J0] has investigated the mapping of ER schemas onto hypercube multiprocessors. The algorithms developed by him map semantically related nodes of the schema graph onto adjacent subcubes of the hypercube architecture. The results obtained by him are of theoretical interest. Nevertheless, they cannot be practically used within our framework. This is because that, in order to maintain the adjacency, a very large number of processing nodes (compared to the number of data cluster groups) will be required, and proper utilization of hardware resources cannot be guaranteed.

We have developed a heuristic algorithm that maps an irregular computation-communication graph onto a set of regularly connected processing nodes, where the number of nodes in the graph equals the number of processing nodes. The heuristic algorithm maps the cluster groups in such a fashion as to reduce the average communication time among any two communicating cluster groups. An estimation is made about the communication cost of the individual cluster groups and a mapping priority is established among the various cluster groups based on the estimated communication cost. Subsequently, the cluster groups are spirally mapped to the

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processing nodes of the network. The mapping is guided by the obtained priority. We have analyzed the performance of the two basic search strategies, namely, the depth-first and the breadth-first, for ordering the mapping of the cluster groups of the computation-communication graph. The following

illustrates the mapping technique. The mapping of the

computation-communication graph of Figure 4.6 across a torus connected set of processing nodes is also shown as an example. Let the computation-communication graph be G(C) = (V(C), E(C), W(C)).

V(C) is a set of vertices representing the time for retrieving and processing the data pertaining to the various data cluster groups obtained after the load balancing phase above.

E(C) a V(C) X V(C) is a set of directed edges (V(Ci), V(Cj)), where (V(Ci),V(Cj) c V(C)), originating from V(Ci) and ending at V(Cj). The edge (V(Ci) ,V(Cj)) represents the

communication of data from the data cluster group represented by V(Ci) to the data cluster represented by V(Cj). Also, if (V(Ci), V(Cj)) e E(C), then (V(Cj), V(Ci)) e E(C). W(C) is a set of weights associated with each of the directed edges of the set of edges E(C). A weight W(i,j) associated with a directed edge (V(Ci), V(Cj)) represents the intensity of data communicated from the cluster group represented by V(Ci) to the cluster group represented by V(Cj). It should be noted that W(j,i) could be different from W(i,j). Let the processor graph be G(P) = (V(P), E(P)).

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V(P) is the set of vertices representing the processing nodes in the parallel processing system. E(P) a V(P) X V(P) is a set of directed edges (V(Pk), V(Pl)), where ( (V(Pk), V(Pl)) c E(P)), originating at V(Pk) and

ending at V(Pl). The edge (V(Pk), V(Pl)) represents the communication link between the processors V(Pk) and V(Pl). It should be noted that in a homogeneous system all the communication links have similar data bandwidth. Also, if (V(Pk), V(Pl)) c E(P), then (V(Pl), V(Pk)) c E(P). The mapping M: V(C) V(P) is one-to-one and is such that the average communication delay among any two processing nodes V(Pk) and V(Pl) mapping the cluster groups V(Ci) and V(Cj) (i.e. M(V(Ci)) = V(Pk), and M(V(Cj)) = V(Pl)) is minimized. The average communication delay among the processing nodes is Avg-Comm-Delay = (Sum-Max-Comm-Delay)/(Sum-Max-Weights)

Sum-Max-Comm-Delay is the sum of the maximum communication delay among all pairs of processors corresponding to the pairs of communicating cluster groups, and Sum-Max-Weights is the sum of the maximal weights among all pairs of communicating cluster groups. Sum-Max-Comm-Delay = Z [ V (V(Ci), V(Cj)) AND (V(Cj), V(Ci)) e E(C)J

(Maximum(W(ij)(V(Ci), V(Cj)), W(j,i)(V(Cj), V(Ci)) ) D(P) (M(V(Ci)) M(V(Cj))) )

D(P) (V(Pk) V(Pl)) is the shortest distance from V(Pk) to V(Pl) in number of hops.

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Sum-Max-Weights = E [V (V(Ci), V(Cj)) and (V(Cj), V(Ci)) c EC]

(Maximum(W(i,j)(V(Ci), V(Cj)), W(j,i)(V(Cj), V(Ci)))

The maximal communication cost along either direction among all pairs of communicating cluster groups is considered. This is because the links connecting the corresponding processing nodes have the same bandwidth, and the data with the maximal size flowing among the two cluster groups in either direction imposes the greater demand on the communication links. The following is an algorithmic

description of the heuristic technique. The first step of the algorithm estimates the communication cost associated with the vertices of the computation-communication graph and assigns mapping priorities to the vertices. The second step of the algorithm maps the individual vertices of the computationcommunication graph onto the vertices of the processor graph. Step 1: Estimate the communication cost associated with each vertex of the graph G(C) and assign mapping priorities to the vertices.

Step 1.1: Estimate the weights associated with each edge of the computation-communication graph.

The weight W(i,j) associated with the flow of data from the cluster group represented by the vertex V(Ci) to the cluster group represented by the vertex V(Cj) is proportional to Data-Trans(V(Ci) ,V(Cj)). The formula for the estimation of Data-Trans is shown in the second step of the grouping phase of Section 4.2.2.

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Step 1.2: Estimate the communication cost associated with each vertex of the computation-communication graph.

As stated above, the communication cost associated with two communicating vertices V(Ci) and V(Cj) is Comm-Cost(V(Ci) ,V(Cj)) = Maximum (Data-Trans(V(Ci) ,V(Cj)), Data-Trans(V(Cj) ,V(Ci)))

Let the communication cost associated with the vertex V(Ci) due to the communication of data with other connected vertices be Comm-Cost(V(Ci)).

Comm-Cost(V(Ci)) = [ [V (V(Ci), V(Ck)) c E(C)] Comm-Cost(V(Ci) ,V(Ck)).

Step 1.3: Sort the vertices of the graph G(C) in the descending order of their communication cost.

Different sorting algorithms can be used with varying complexities to perform the sorting. The following pseudo code illustrates the sorting procedure using one of the simplest sorting algorithms, namely, the bubble sort.

Let the assignment priority of the vertices of the computation-communication graph be stored in the array named Priority. Initially, the priority among the vertices of the computation-communication graph is arbitrarily assigned. For i = 1 To Number of vertices in V(C)

Priority[i] = V(Ci)

For i = 1 To (Number of vertices in V(C) 1)

For j = (i + 1)'To Number of vertices in V(C)

If (Comm-Cost(Priority[i]) < Comm-Cost (Priority[j])), then

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Swap (Priority[i], Priority[j])

The weights associated with the individual edges of the example computation-communication graph are shown in Figure 4.6. The communication cost, in thousands of object instances communicated, of the individual vertices of the example computation-communication graph are tabulated in Table 4.9. The vertices are sorted and presented in the order of their assignment priorities.

Step 2: Map the vertices of the vertex set V(C) onto the vertices of the vertex set V(P) using the priority established in step 1.

Let the set of vertices of V(C), mapped to the vertices of V(P), at any instant of the mapping process be AssignedSet(C).

Let the set of vertices of V(P) that have been assigned the vertices of V(C) be Assigned-Set(P). Let the vertex of V(C) currently being mapped to a vertex of V(P) at any instant be Current-Vertex(C). Let the vertex of V(P) currently being assigned the vertex of V(C) be Current-Vertex(P).

Initially, Assigned-Set(C) = Nil. Assigned-Set(P) = Nil.

Step 2.1: Assign the vertex in V(C) with the highest priority onto any vertex of V(P).

Since the vertices of V(P) are regularly connected any vertex can be chosen for the initial assignment. The vertex,

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among the vertices of V(C), with the highest priority has the maximum communication associated with it and is mapped first. The possibility of mapping the connected vertices, of the computation-communication graph, across the processor vertices that are closely connected is higher at the initial stages of the mapping process. The vertices of the computationcommunication graph with higher communication requirements are mapped before the vertices with lower communication requirements. This enables the reduction in the average communication delay among communicating data clusters. Current-Vertex(C) = Priority[l]. Current-Vertex(P) = V(P1).

M(Current-Vertex(C)) = Current-Vertex(P). Assigned-Set(C) = Assigned-Set(C) + Current-Vertex(C). Assigned-Set(P) = Assigned-Set(P) + Current-Vertex(P). Step 2.2: Determine the next vertex of the graph G(C) to be mapped.

Rooting at the vertex with the highest communication cost, the subsequent vertices of the graph G(C), to be mapped, are determined by searching the other connected vertices in the graph. We have analyzed the performance of the two basic search techniques, namely, the depth-first, and the breadth-first. The search techniques are described below.

4.2.3.1. Depth-first search of the Qraph G(C)

Using this search technique, the vertices of the computation-communication graph G(C) are navigated in the

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depth-first fashion rooting at the vertex with the highest communication cost. Starting from the Current-Vertex(C), the last vertex of G(C) that was mapped, the next vertex is determined by first navigating the immediate connected vertices of the Current-Vertex(C). Among the immediate

connected vertices, the one with the maximum communication associated with it is chosen. If all the immediate connected vertices of the Current-vertex(C) are already mapped, then the current vertex is backtracked to the immediate ancestor of the Current-vertex(C) in the depth-first spanning tree of G(C) and the ancestor's immediate connected vertices are navigated and analyzed. The backtracking to the ancestors is recursively performed until an unmapped vertex is found. Let the set of vertices connected to the Current-Vertex(C) and not yet assigned be Connected-Set(C). Let the vertex, among the vertices in the Connected-Set(C), with the highest assignment priority be High-Pri-Vertex(C). Let PARENT be a function than returns the immediate ancestor vertex of any vertex in the depth-first spanning tree of the graph G(C).

The following pseudo code illustrates the process. Initially, Connected-Set(C) = Nil. Found = FALSE

REPEAT

(Y (Current-Vertex(C),V(Cj)) e E(C))

If (V(Cj) $ Assigned-Set(C)), then

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Connected-Set(C) = Connected-Set(C) + V(Cj)

If (Connected-Set(C) = Nil), then

Current-Vertex(C) = PARENT(Current-Vertex(C))

Else

Found = TRUE

UNTIL Found
i= 1

Found = FALSE

REPEAT

i= i+ 1

If (Priority[i] c Connected-Set(C)), then

Found = True

UNTIL Found.

High-Pri-Vertex(C) = Priority[i] Current-Vertex(C) = High-Pri-Vertex(C).

Figure 4.7(a) shows the depth-first mapping tree of the computation-communication graph shown in Figure 4.6. Vertex V(C5) has the highest mapping priority and is mapped first. Among the immediately connected vertices of V(C5), the vertex V(C4) has the highest mapping priority and is mapped next. Vertex V(C7) is mapped after the mapping of V(C4) since it has the highest mapping priority among the unmapped immediately connected vertices of V(C4). Subsequently, vertices V(C3), V(C8), and V(C9) are mapped. Since the immediately connected vertex of V(C9) is already mapped, the search is backtracked and the connected vertices of V(C8) are searched. The

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connected vertices of V(C8) are also mapped and the search is backtracked to the unmapped connected vertices of V (C3) Among the unmapped connected vertices of V(C3), the vertex V(Cl) has the highest mapping priority and is mapped next. Since all the connected vertices of V(Cl) are mapped, the search is again backtracked to the connected vertices of V(C3) and the unmapped vertex with the highest mapping priority, namely, vertex V (C2), is mapped next. Subsequently, vertex V(C6) is mapped.

4.2.3.2. Breadth-first search of the graph G(C)

Using this mapping strategy, the vertices of G(C) are navigated and mapped in a breadth-first fashion rooting from

the vertex that has the maximal communication cost. After the mapping of the root vertex, the vertices at the first level of the breath-first tree are determined by ordering all the immediately connected vertices of the root vertex. The

vertices are ordered and mapped in the descending order of their assignment priorities. once all the vertices at the

first level are mapped, the unmapped immediately connected vertices of the each of individual vertices of the first level are sorted and mapped in the descending order of their assignment priorities. The process is repeated until all the vertices of the graph G(C) are mapped. The following pseudo code illustrates the step.

Let the current level of the breadth-first tree be CurrentLevel.

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Let the array storing the vertices at the current level sorted in the order of their mapping be Current-Level-Array. Let the total number of vertices in the current level be NCurrent-Level.

Let the number of unmapped vertices at the current level be NUnmapped.

Let the array storing the vertices, at the level higher than the current level, sorted in the order of their mapping be Next-Level-Array.

Let the total number of vertices at the higher level be NNext-Level.

Initially, the state of the variables and the arrays will be as follows:

N-Current-Level = 0,

N-Next-Level = 0,

N-Unmapped = 0, and

Current-Level-Array and Next-Level-Array will have no vertices.

During the first execution of this step, the vertices at the first level of the breadth-first mapping tree are obtained by searching all the unmapped connected vertices of the root vertex determined in step 2.1. The vertices are ordered based on the communication cost associated with them. The vertex, at the first level, with the highest communication cost is returned as the next vertex to be mapped. The following

pseudo code illustrates the process.

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If (N-Unmapped = 0) AND (Current-Level = 0), then

Current-Level = Current-Level + 1

(Y (Current-Vertex(C), V(i)) e E(C))

If (V(i) $ Assigned-Set(C)), then

N-Unmapped = N-Unmapped + 1

Current-Level-Array[N-Unmapped] = V(i)

For j = 1 To (N-Unmapped 1)

For k = j To N-Unmapped

If (Comm-Cost(Current-Level-Array[j]) >

Comm-Cost(Current-Level-Array[k])), then

Swap (Current-Level-Array[ j ,Current-Level-Array[k])

Current-Array-Length = N-Unmapped

Current-Vertex(C) = Current-Level-Array[N-Unmapped]

N-Unmapped = N-Unmapped 1

In subsequent executions of this step, if the CurrentLevel-Array contains unmapped vertices (i.e., N-Unmapped 0), the unmapped vertex, in the current level, with the highest communication cost is returned as the next vertex to be mapped. The following pseudo code illustrates the process.



If (N-Unmapped o 0), then

Current-Vertex(C) = Current-Level-Array(N-Unmapped

N-Unmapped = N-Unmapped 1

However, if all the vertices at the current level are mapped, the unmapped vertices at the next level are searched. The unmapped vertices which are immediately connected to each

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of the vertex at the current level are found. The connected vertices of each vertex, at the current level, are ordered based on their communication costs. The connected vertices of the vertex at the current level that has the maximal communication cost are mapped first. Among the connected vertices, at the next level, the vertex with the maximal communication cost is returned as the next vertex to be mapped. The following pseudo code illustrates the step. Let Current-Vertex-Limit be an index of the Next-Level-Array. If (N-Unmapped = 0) AND (Current-Level 0 0), then

N-Next-Level = 0

Current-Vertex-Limit = 1

For i = 1 To N-Current-Level

(V (Current-Level-Array[i], V(j)) e E(C))

If (V(j) i Assigned-Set(C)), then

N-Next-Level = N-Next-Level + 1

Next-Level-Array(N-Next-Level] = V(j)

For k = Current-Vertex-Limit To (N-Next-Level 1)

For 1 = k To N-Next-Level

If (Comm-Cost(Next-Level-Array[k]) >

Comm-Cost(Next-Level-Array[l])), then

Swap (Next-Level-Array[k], Next-Level-Array[l))

Current-Vertex-Limit = N-Next-Level + 1

For i = 1 TO N-Next-Level

Current-Level-Array(i] = Next-Level-Array(i]

N-Unmapped = N-Next-Level

..








N-Current-Level = N-Next-Level

Current-Vertex(C) = Current-Level-Array[N-Unmapped]

N-Unmapped = N-Unmapped 1

Figure 4.7(b) shows the breadth-first mapping tree of the computation-communication graph of Figure 4.6. Vertex V(C5) has the highest priority of mapping and is mapped first. All the connected vertices of V(C5) are mapped in the next level. The connected vertices are mapped in the order of their assignment priorities. For example, vertex V(C4) has a higher assignment priority compared with vertices V(Cl), V(C2), V(C7), and V(C8) and is mapped after the vertex V(C5). Vertex V(C8) has the second highest assignment priority among the connected vertices of V(C5) and is mapped next. Similarly, the subsequent mapping of vertices V(C7), V(Cl), and V(C2) can be explained. Next, the unmapped immediately connected vertices of V(C4) are mapped. The connected vertices are mapped in the order of their assignment priorities. Thus, vertex V(C3) is mapped before the mapping of vertex V(C6). Subsequently, vertex V(C9), the unmapped connected vertex of the vertex V(C8) is mapped. Since no more vertices remain unmapped the mapping process is completed. Step 4.3: Determine the next vertex of the processor graph G(P) on which the Current-Vertex(C) has to be mapped and map the Current-Vertex(C). If all the vertices of the graph G(C) are not mapped, then repeat the mapping process.

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The Current-Vertex(C) is mapped onto the vertex CurrentVertex(P) of G(P) such that the distance between the CurrentVertex(P) and the vertex of G(P) mapping the immediate ancestor of Current-Vertex(C) is as small as possible. Thus, at the first level, all the immediately connected vertices of the vertex mapping the parent of the Current-Vertex(C) are analyzed. Current-Vertex(C) is mapped onto any one of the nonmapped vertex among the immediately connected vertices. If all the vertices at a distance of one hop (i.e., all the immediately connected vertices) are found mapped, the vertices at a distance of two hops are analyzed, and Current-Vertex(C) is mapped onto any nonmapped vertex among them. The

immediately connected vertices of all the vertices at a distance of one hop are at a distance of two hops. All the immediately connected vertices of the vertex mapping the parent of the Current-Vertex(C) are ordered based on the communication cost associated with the vertices of G(C) mapped on them. The neighbors of a connected vertex mapping a vertex of G(C) with lower communication cost are analyzed before analyzing the connected vertex mapping a vertex with higher communication cost. The mapping on the neighbors of vertices with least communication cost enables the vertices of G(C) with higher communication costs to be mapped across the vertices of G(P) as close as possible. Vertices at a further or longer distance from the vertex corresponding to the parent of the Current-Vertex(C) are analyzed when all the vertices at

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a shorter distance are already mapped. The following pseudo code illustrates the step.

The vertex of the graph G(P) corresponding to the immediate ancestor of the Current-Vertex(C) is Parent-Current-Vertex(P) = INVERSE M (PARENT (Current-Vertex(C)))

Let the set of vertices connected to the Current-Vertex(P) be Connected-Set(P).

Starting at the first level, the immediately connected vertices of Parent-Current-Vertex(P) are analyzed. Subsequently at higher levels the vertices at higher distances are analyzed.

Let the array storing the set of vertices of G(P), whose neighbors are being analyzed at any instant in a sorted order based on the communication costs associated with the vertices of G(C) mapped on them, be Current-Level-Array. Let the number of vertices in the Current-Level-Array be NCurrent-Level.

Let the number of vertices immediately connected to the vertex in the ith element of the N-Current-Level be N-ConnVertices(i).

Let the array storing all the neighboring vertices of the vertices in the Current-Level-Array be Next-Level-Array. Let the number of vertices in the Next-Level-Array be N-NextLevel.

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Initially, Current-Level-Array has only one vertex, namely the Parent-Current-Vertex(P). Thus, N-Current-Level = 1 and Current-Level-Array[l] = Parent-Current-Vertex(P). N-Next-Level = 0 Found = FALSE

REPEAT



REPEAT

Current-Vertex-Limit = 1

V (Current-Level-Array(i], V(Pj)) c E(P)

N-Next-level = N-Next-Level + 1

Next-Level-Array(N-Next-Level] = V(Pj)

k = N-Next-Level + 1

REPEAT

k = k 1

If (Next-Level-Array[k] t Assigned-Set(P)), then

Found = TRUE

UNTIL (Found = TRUE) OR (k = 1)

If (Found = TRUE), then

Current-Vertex(P) = Next-Level-Array[k]

If (Found = FALSE), then

FOR k = Current-Vertex-Limit TO (N-Next-Level 1)

FOR 1 = k + 1 TO N-Next-Level

If (Comm-Cost(INVERSE M(Next-Level-Array[k]) >

Comm-Cost(INVERSE M(Next-Level-Array[1])), then

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Temporary-Storage = Next-Level-Array(k]

Next-Level-Array[k] = Next-Level-Array[l]

Next-Level-Array[l] = Temporary-Storage

Current-Vertex-Limit = N-Next-Level + 1

i= i+ 1

UNTIL (Found = TRUE) OR (i = 0)

If (FOUND = FALSE), then

FOR k = 1 TO N-Next-Level

Current-Level-Array[k) = Next-Level-Array(k]

N-Current-Level = N-Next-Level UNTIL (Found = TRUE)

Figure 4.8(a) shows the mapping of the depth-first tree of Figure 4.7(a) onto the vertices of the processor graph. The vertices of the processor graph are connected in the form of a torus. Vertex V(C5) is first mapped onto the processor vertex V(Pl). Since the vertices of the processor graph are connected in a regular and homogeneous fashion, any vertex can be chosen for the initial assignment. Vertex V(C4) is next mapped onto the processor vertex V(P7) since vertex V(P7) is directly connected to the vertex V(P1). Similarly, vertices V(C7), V(C3), V(C8), V(C9), and V(Cl) are mapped onto the processor vertices V(P4), V(P5), V(P2), V(P8), and V(P6) respectively. Since all the directly connected vertices of V(P5), the vertex corresponding to the vertex V(C3), are already mapped, the vertex V(C2) is mapped onto a vertex that is at a distance of two hops from V(P5). Vertex V(C2) is

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