<%BANNER%>

Topological Reasoning between Complex Regions in Databases with Frequent Updates

Permanent Link: http://ufdc.ufl.edu/UFE0042176/00001

Material Information

Title: Topological Reasoning between Complex Regions in Databases with Frequent Updates
Physical Description: 1 online resource (40 p.)
Language: english
Creator: Khan, Md
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: geographic, spatial, topological
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Reasoning about space has been a considerable field of study both in Artificial Intelligence and in spatial information theory. Many applications benefit from the inference of new knowledge about the spatial relationships between spatial objects on the basis of already available and explicit spatial relationship knowledge that we call spatial (relationship) facts. Hence, the task is to derive new spatial facts from known spatial facts. A considerable amount of work has focused on reasoning about topological relationships (as a special and important subset of spatial relationships) between simple spatial objects like simple regions. There is a common consensus in the GIS and spatial database communities that simple regions are insufficient to model spatial reality and that complex region objects are needed that allow multiple components and holes. Models for topological relationships between complex regions have already been developed. Hence, as the next logical step, the goal of this thesis is to develop a reasoning model for them. Further no reasoning model considers changes of the spatial fact basis stored in a database in between the queries. We show that conventional modeling suffers performance degradation when the database is frequently changing. Our model does not assume any geometric representation model or data structure for the regions. The model is also backward compatible which means that it is also applicable to simple regions.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Md Khan.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Schneider, Markus.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042176:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042176/00001

Material Information

Title: Topological Reasoning between Complex Regions in Databases with Frequent Updates
Physical Description: 1 online resource (40 p.)
Language: english
Creator: Khan, Md
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: geographic, spatial, topological
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Reasoning about space has been a considerable field of study both in Artificial Intelligence and in spatial information theory. Many applications benefit from the inference of new knowledge about the spatial relationships between spatial objects on the basis of already available and explicit spatial relationship knowledge that we call spatial (relationship) facts. Hence, the task is to derive new spatial facts from known spatial facts. A considerable amount of work has focused on reasoning about topological relationships (as a special and important subset of spatial relationships) between simple spatial objects like simple regions. There is a common consensus in the GIS and spatial database communities that simple regions are insufficient to model spatial reality and that complex region objects are needed that allow multiple components and holes. Models for topological relationships between complex regions have already been developed. Hence, as the next logical step, the goal of this thesis is to develop a reasoning model for them. Further no reasoning model considers changes of the spatial fact basis stored in a database in between the queries. We show that conventional modeling suffers performance degradation when the database is frequently changing. Our model does not assume any geometric representation model or data structure for the regions. The model is also backward compatible which means that it is also applicable to simple regions.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Md Khan.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Schneider, Markus.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042176:00001


This item has the following downloads:


Full Text





TOPOLOGICAL REASONING BETWEEN COMPLEX REGIONS IN DATABASES WITH
FREQUENT UPDATES


















By

MD ARIFUL HASAN KHAN


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2010































@ 2010 Md Ariful Hasan Khan
































To my parents









ACKNOWLEDGMENTS

First of all, I thank Dr. Markus Schneider for his invaluable guidance and

encouragement. Without his guidance this thesis would not have been possible. I

am also grateful to my supervisory committee members Dr. Jonathan Liu and Dr. Alin

Dobra suggestions and feedbacks. I am extremely fortunate for having such a loving

and caring parents. Their words and support have been the main motivating factor all

through my education.









TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... .............. 4

LIST OFTABLES ................... ... ................ 6

LIST OF FIGURES ................... .. ................ 7

ABSTRACT. ...................... .................. 8

CHAPTER

1 INTRODUCTION ................... .. .............. 9

2 BACKGROUND AND RELATED WORK ................ ...... 11

2.1 Background . .. 11
2.1.1 Spatial Objects ................... .......... 11
2.1.2 Spatial Relationships ..... ........ 11
2.1.3 Topological Reasoning with Complex Regions ........... 13
2.2 Related Work .................... .............. 15

3 LOCALINFERENCE ..................... ............ 17

3.1 Set Relationships between the Interiors ................... 17
3.2 Inference Rules . .. 19
3.3 Relationship Identifying Process ..... .... 25

4 GLOBAL INFERENCE ................... ............. 29

4.1 Global Inference in Databases with Frequent Updates ... 29
4.2 An Algorithm for Reasoning between Complex Regions ... 33
4.3 Simulation and Results ................. ......... 34

5 CONCLUSIONS AND FUTURE WORKS ..................... 37

REFERENCES ....................... ................ 38

BIOGRAPHICAL SKETCH .................. ............ 40









LIST OF TABLES


Table page

2-1 Number of Topological Predicates Between Two Complex Spatial Objects. 13

3-1 33 possible topological relationships between two complex regions. 26









LIST OF FIGURES


Figure page

2-1 Examples of a (a) complex point object, (b) a complex line object, and (c) a
com plex region object .. . 12

2-2 Eight basic topological relationships between two simple regions. ... 12

3-1 Steps of Local Inference. ................... .......... 17

3-2 (a) A complex region with its faces and holes, and (b) its interior, boundary,
and exterior .................. ................. 18

3-3 (a) 9-Intersection Matrix, (b) complex regions A and B meet, (c) Rmeet(A, B). 18

3-4 The interiors of A and C: (a) intersects, (b) does not intersect. ... 21

3-5 Decision tree of the relationship space for complex regions. ... 27

3-6 The algorithm IdentifyRelationship.. ....................... .28

4-1 A chain of relationships.. ............................ 31

4-2 Multiple chains of relationships. ... 32

4-3 The algorithm ReasoningBetweenComplexRegion. ... 33

4-4 Performance of the heuristic for different database sizes. ... 35









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

TOPOLOGICAL REASONING BETWEEN COMPLEX REGIONS IN DATABASES WITH
FREQUENT UPDATES

By
Md Ariful Hasan Khan

August 2010

Chair: Markus Schneider
Major: Computer Engineering

Reasoning about space has been a considerable field of study both in Artificial

Intelligence and in spatial information theory. Many applications benefit from the

inference of new knowledge about the spatial relationships between spatial objects on

the basis of already available and explicit spatial relationship knowledge that we call

spatial (relationship) facts. Hence, the task is to derive new spatial facts from known

spatial facts. A considerable amount of work has focused on reasoning about topological

relationships (as a special and important subset of spatial relationships) between

simple spatial objects like simple regions. There is a common consensus in the GIS

and spatial database communities that simple regions are insufficient to model spatial

reality and that complex region objects are needed that allow multiple components

and holes. Models for topological relationships between complex regions have already

been developed. Hence, as the next logical step, the goal of this thesis is to develop a

reasoning model for them. Further no reasoning model considers changes of the spatial

fact basis stored in a database in between the queries. We show that conventional

modeling suffers performance degradation when the database is frequently changing.

Our model does not assume any geometric representation model or data structure

for the regions. The model is also backward compatible which means that it is also

applicable to simple regions.









CHAPTER 1
INTRODUCTION

Understanding the topological relationships between objects in space has become

a multidisciplinary research issue involving Al, CAD/CAM systems, cognitive science,

computer vision, image databases, linguistics, robotics, GIS, and spatial databases.

From a spatial database and GIS point of view, topological relationships are necessary

as filter conditions for spatial selections and spatial joins as well as for spatial data

retrieval and analysis. In spatial databases and GIS, we generally deal with a large

number of spatial objects. Hence, it is not uncommon that we do not have all possible

relationships available between every pair of spatial objects all the time. This situation

can arise either due to a lack of information or since it is impossible to get all the

relationships. To deal with this problem of a lack of complete knowledge, we need a

process through which we can infer the topological relationship between two spatial

objects where the relationship does not currently exist in the knowledge base. This

process is called reasoning. Hence, reasoning about topological relationship is a

method of inferring new topological relationships, call 2D spatial facts, between two

spatial objects using the other existing spatial facts in the knowledge base. For example,

given three objects A, B and C, and given two topological relationships Rx(A, B) and

Ry(B, C), reasoning helps us to find out the relationship Rz between A and C where

Rz does not exist in the knowledge base. This process is also called the composition of

relationships which is the most common method of reasoning.

So far, the main focus of the available reasoning models is to deal with simple

regions. But in the real world we often face the situation where a real objects cannot be

represented by simple regions alone. For example, Italy contains the Vatican as a hole,

and the Galapagos island does not consist of a single island but rather of a collection of

many islands. These spatial phenomena cannot be represented by simple regions. The

second problem is that the current reasoning models hardly take the changes of spatial









facts into account. It is natural that often the information is added, deleted or updated in

the databases. So it is important to understand as well as to consider the effect of such

changes while designing a reasoning model.

The main goal of this thesis is to develop a reasoning model for complex regions.

The main challenge is to deal with a large number of possible topological relationships

between two complex regions as well as to deal with a large number of such regions.

Our second goal is to derive a set of inference rules by which the inference of relationships

is performed. Since the type for simple regions is a subset of the type for complex

regions, it is also our goal that the reasoning model is able to handle simple regions

without requiring any modification. Finally, we show the effect of the changes of spatial

facts on the reasoning process, and we propose an algorithm to handle those changes.

We propose a generalized process to infer new relationships between complex

regions which is not restricted by the number of regions as well as changes in the

database. The process has two basic steps. In the first step, we perform the reasoning

process involving three regions, call it Local Inference. In the second step, we extend

this local inference to N regions and hence call it Global Inference.

The remainder of the thesis is organized as follows: Section 2 discusses background

and related work regarding reasoning models. In Sections 3 and 4, we describe the local

and global inference respectively. Finally, Section 5 draws some conclusions and

discusses future work.









CHAPTER 2
BACKGROUND AND RELATED WORK

2.1 Background

In this section we discuss about the different types of spatial objects followed by

the different types of relationships between spatial objects. Then, We explain the basic

steps of the reasoning process and the general algorithms to implement these steps.

2.1.1 Spatial Objects

In the past, numerous data models have been proposed with the aim of formulating

spatial objects in databases and GIS. Spatial objects embedded in 2D-space can be of

three types: (i) point objects, (ii) line objects, and (iii) region objects. Point objects are

0-dimensional spatial objects and only provide positions. Line objects are 1-dimensional

linear spatial objects that have a length. Region objects are 2-dimensional spatial

objects with an extent (i.e., both height and width). Each kind of spatial object can be

categorized as either a simple spatial object [1-4] or a complex spatial object [5-7].

In this thesis we mainly consider complex region objects. The Figure 2-1 shows the

different types of complex spatial objects. A simple region is topologically equivalent

to a closed disc; it does not have holes. However, a complex region may have multiple

components, call faces and may have multiple holes. One important aspect is that for

the reasoning process the spatial objects are only needed as symbolic terms; their

geometries are not required. The mathematical basis for formalizing the spatial objects

(i.e., both simple and complex) is point set theory and point set topology which assumes

that the planar space is comprised of an infinite number of points.

2.1.2 Spatial Relationships

There are three kinds of spatial relationships: (i) directional relationships, (ii)

topological relationships and (iii) distance relationships. Directional relationships validate

the cardinal direction between two spatial objects (e.g., north, southwest). Distance

relationships validate the qualitative distance between spatial objects (e.g., far, close).















(a) (b) (c)
Figure 2-1. Examples of a (a) complex point object, (b) a complex line object, and (c) a
complex region object.


X disjoint Y


X meet Y


X covered by Y


X

X Y X Y
Y

X overlap Y X equal Y X inside Y

Figure 2-2. Eight basic topological relationships between


Y X



X contain Y

two simple regions.


Our focus is on topological relationships which characterize the relative position of

two spatial objects (e.g., overlap, meet). For example, the eight basic topological

relationships between two simple regions are shown in the Figure 2-2 .An important

approach for characterizing the topological relationships between spatial objects is

known as 9-intersection model [8]. By using this model, the authors in [7] have identified

the topological relationships between any two complex spatial objects irrespective of

their types. Thirty three topological relationships have been found for two complex

regions. The following Table 2-1 shows the number topological relationships possible

between every combination of spatial objects.


X covers Y









Table 2-1. Number of Topological Predicates Between Two Complex Spatial Objects.
Complex Point Complex Line Complex Region
Complex Point 5 14 7
Complex Line 14 82 43
Complex Region 7 43 33

2.1.3 Topological Reasoning with Complex Regions

As mentioned, reasoning models which can only deal with simple regions are

not enough to represent real world scenarios. For example, some biologists who

are researching on the Darwin's theory, are looking for a possible evolutionary link

between a land species X and an amphibian species Y around the Galapagos islands.

The hunting areas of the species X and Y are the regions A and C respectively

and the Galapagos islands is the region B. The relationships between the hunting

areas of the species X and Y with the Galapagos islands are Rx(A, B) and Ry(B, C)

respectively. Having these informations, the biologist may be able to get a possible link

between these two species by looking at the relationship Rz(A, C) (i.e., the topological

relationship between their hunting areas) through the reasoning process. Now the

regions in question are: the Galapagos islands which consists of many islands (i.e.,

complex region) and the living areas of the species may confined to one island (i.e.,

simple regions) or may extend to many of these islands (i.e., complex regions) or

may have a lake inside it (e.g., region with holes i.e., complex region). Hence, we can

see from the example that the regions A, B and C can be all complex regions or any

combination of simple and complex regions. Above scenario can easily be extended

from three regions to N regions. Therefore, a more generalized and comprehensive

reasoning model is required.

The first step of the reasoning process is the local inference involving three regions

in the form of Rx(A, B) and Ry(B, C). Here, Rx and Ry are the spatial facts between

the complex regions A, B and B, C respectively. The goal is to find the relationship









Rz(A, C). This local inference is carried out by a process called composition of relation-
ships by means of a set of inference rules. It is important to note that the composition

of relationships does not depend on the extent or the spatial characteristics of the

regions in concern. Therefore, the composition of relationships can be denoted as

Rx o Ry = Rz. Local inference alone is not enough for inferring relationships between
two complex regions. Consider the chain R1(A, B), R2(B, C), R3(C, D), R4(D, E) of

topological relationships among the five regions A, B, C, D, and E. In this situation,

local inference alone is not sufficient to infer the relationship between A and E. Because

an intermediate object is required that is in relationship to both A and E. In this example

scenario, such an intermediate region does not exist. Thus, the global inference comes

into play which makes use of the composition of relationships to infer relationships

between any two regions in the knowledge base.

An important observation is that the global inference is orthogonal to the local

inference. That is, global inference can employ any algorithm to infer relationships

globally as long as the composition of relationships is available. Unsurprisingly, the

global inference is a constraint satisfaction problem. A constraint satisfaction problem

(CSP) is defined as a triple (X, D, C), where X is a set of variables, D is a domain of

values, and C is a set of constraints. Every constraint is in turn a pair (t, R), where

t is a tuple of variables and R is a relation. The CSP can be viewed as a directed

graph, where the nodes are the variables and the edges between two variables are

the relations or the constraints. This directed graph is also called constraint network.

In our case, the relations are all binary topological relationships and the variables are

spatial objects (i.e., regions); we call this graph representation binary spatial constraint

network (BSCN). The class of algorithms for global inferencing by using BSCN is based

on a path consistency procedure. A pair of variables is path consistent with a third

variable if each consistent evaluation of the pair can be extended to the other variable in

such a way that all binary constraints are satisfied. Formally, the variables A and C are









path consistent with B if there is a relation R1(A, C) that satisfies the binary constraint

between A and C and if there are two relations R2(A, B) and R3(B, C) that satisfy the

constraint between A and B and between B and C, respectively. A simple observation

tells us that path consistency can be achieved through composition of relationships.

The algorithm applies the path consistency procedure over all combinations of nodes in

the BSCN until no new relationships can be inferred. An important point is that, given a

partially observed knowledge base, the path consistency algorithms derive the complete

knowledge, i.e., the relationships between every pair of objects. That is, after running

the global inference algorithm the knowledge base becomes complete and it takes 0(1)

time to find the relationships) between any pair of the complex regions.

2.2 Related Work

Numerous studies have been done on topological relationships as well as

topological reasoning. The reasoning process tries to infer the relationships which

are defined and derived by the relationship model. Therefore, reasoning models are

dependent on the underlying relationships models. Researchers from different domains

such as Al, mathematics, GIS and databases, have been contributing to this field of

study. The authors of the papers [9-11] attack this problem with the algebraic logic

approaches. The authors of the paper [8] defined spatial objects based on topological

set theory and proposed 9-Intersection Model as a way to characterize the spatial

objects. Based on the topological set theory, the authors proposed the reasoning

models about simple regions [1, 11-13], simple regions with holes [14, 15]. In [16] the

authors propose a reasoning model taking the concavity of the regions into the account

along with their convex hulls. Most of the times, the inferred relationship between

spatial objects may not be unique, i.e., the inferred relationship can be a disjunction of

several basic relationships. Based on this observation, the authors of [17, 18] propose

hierarchical models for topological reasoning.









All of the above mentioned studies mainly focused on the local inference (i.e.,

composition of relationships involving three objects by means of inference rules). It

is well understood that local inference is an essential and basic step of the reasoning

process but without global inference the process is not complete. The reason behind

more focus on local inference is because global inference is a constraint satisfaction

problem(CSP) [19-22] which is an extensively studied topic and is independent of

the local inference process. The authors of [10, 23, 24] studied the issues related to

constraint satisfaction for spatial objects such as the complexity and the tractability.

So far, the lowest complexity O(n3) of CSP algorithms is proposed by the authors

in [20-22]. All of these CSP algorithms operate on the static knowledge base. That is,

given a BSCN, the algorithm runs and able to infer relationships between any pair of

complex regions. But over time, the existing facts may change and the CSP algorithms

are not designed to handle changes. To best of our knowledge, none of the reasoning

models deal with the changes of the spatial facts and our work is motivated by this

issue.









CHAPTER 3
LOCAL INFERENCE

Local inference takes two topological relationships(Rx(A, B) and Ry(B, C)),

compose them and infer relationships) Rz(A, C). Since 9-intersection matrix can

uniquely characterize each topological relationship, the inputs of the local inference

can be the two 9-intersection matrices and the output is a set of inferred relationships.

The Figure 3-1 shows the three steps of the local inference process. At first step,

the corresponding set relationships (i.e., subset relationships, empty/nonempty

intersections) between the interiors of the regions are evaluated from the 9-intersection

matrices. Then the inference rules are being applied to find out the 9-intersection

predicate values between A and C. At the last step, the inferred relationships are being

identified from the predicate values.


Set Relationship 9-Intersection
91M(A,B) between AO & B Predicates Values
Inference between A & C Relationship Inferred
RSet Rela p ules Identifying Process Relationship(s)
Set Relationship
91M(B,C) between B& C
Local Inference

Figure 3-1. Steps of Local Inference.


3.1 Set Relationships between the Interiors

According to the point set topology, each spatial object can be characterized

by three mutually exclusive point sets in the topological space R2. These sets are

the interior (AO), the boundary (OA) and the exterior (A-) for any spatial object A

(Figure 3-2b). The 9-intersection model uses nine predicates to check the nine

intersections of these point sets provided by two spatial objects A and B for non-emptiness.

Each topological relationship between any two spatial objects is characterized by a

unique combination of nine Boolean values. The 9-intersection predicates are arranged

in a 9-intersection matrix (Figure 3-3a).














faces (component

boundary
(a) (b)
Figure 3-2. (a) A complex region with its faces and holes, and (b) its interior, boundary,
and exterior.

On the other hand, the interior, boundary, and exterior of a spatial object are
uniquely defined and disjoint from each other [7]. Therefore, according to the regularized
definition of complex regions, it is sufficient to specify any of these three sets to uniquely
characterize a region object. In this document we consider the interior of a complex
region to uniquely characterize it. Hence, for each topological relationships, there is a
set relation between the interiors of the two complex regions. That is, either the interior
of A is a subset or superset or equal or disjoint or overlaps the interior of B. In [8] the
authors showed the way to find out the set relationship between any two components
of a region object from the 9-intersection matrix by using the topological properties of
the spatial regions. We employ that same technique to find out set relation between the
interiors of the two participating regions of a topological relationship.


A n B 0 AO n 9B 0 A n B- 0 __ 0 0 1
B9A n B A n 9B o A n B- 1
A- n B A- n 9B o 0 A- n B- o 1


(a) (b) (c)

Figure 3-3. (a) 9-Intersection Matrix, (b) complex regions A and B meet, (c) Rmeet(A, B).









3.2 Inference Rules

From set theory, two non-empty sets X and Y must have one of the following five

relations: (i) X is a proper subset of Y, (ii) X is equal to Y, (iii) Y is a proper subset of

X, (iv) X and Y have some common and some different elements, and (v) X and Y do

not have any common element. The fourth relation, we call it overlap, denotes that two

sets have common elements but none of them is the proper subset of the other. We

extend these five relations to eight by adding special cases to the relations (i), (iii) and

(v) using the spatial properties. Consider X and Y as the interiors of the two regions A

and B respectively. Then the relation (i) denotes that the region A is completely inside

the region B. There can be two special cases of this scenario and they are: (a) A is

inside B and their boundaries touch and (B) A is inside B and their boundaries do not

touch. Similarly, these two special cases also hold for relation (iii) and (iv).

Let AO and BO denotes the interior sets. The symbols c, A, -, and = have their

usual meaning. The symbol o denotes the predicate for overlap, i.e., A o BO

(AO n BO 0 A A B 0 A BO Ao 0). The predicate for a non-empty

intersection, i.e., Ao n BO 0, is denoted by AoB, and the predicate for an empty

intersection, i.e., Ao n BO = 0, is denoted by -AB. So, the eight relations between the

interiors of two region objects are the following:

1. A C Bo A OAOB

2. A c Bo A OAOB

3. AO = Bo

4. AO o B

5. -AB A OAOB

6. -AB A -OAOB

7. BO c Ao A 9AOB

8. B c Ao A -iAOB









The relations 1 and 2 are two special cases of the original relation (i). Similarly,

the relations 5 and 6 as well as the relations 7 and 8 are special cases of the original

relations (iii) and (v). Unsurprisingly, these five basic and eight extended relations

correspond to the RCC-5 and RCC-8 [4, 24] respectively. Most importantly, these eight

relations hold for any type of region objects (i.e., simple, complex). Because, simple

region is nothing but a single component complex region without any hole. On the

other hand, since we only consider the interior as a whole which means the interior

of a complex region is the union of the interiors of its all faces, it does not matter how

many holes and components are in that complex region. Since these eight relations

completely characterize the relations between the interiors of two complex regions, any

relationship between two complex regions A and B must include exactly one of these

relations. Therefore, if we have Rx(A, B) and Ry(B, C) then by the transitivity property,

the interiors of A and C must belong to exactly one of the 8 x 8 = 64 configurations

of these relations. That is, for each relation between A and B, there are eight possible

relations between B and C which gives us 64 configurations.

For each of these 64 configurations, we determine the 9-intersection predicate

values between A and C. As an example, for the configuration AO c B0 A -'A9B and

B c Co A -i9BOC by applying simple set theory logics we get, AO c B0 A BO c

C = A0 c Co = A0 n Co 0. That means for this configuration of A, B and

B, C the interior-interior intersection between A, C is always true. Similarly, for the

same configuration we can prove that the interior-exterior intersection between A

and C is always false. We know that the three components (i.e., interior, exterior and

boundary) of a region object are mutually exclusive (i.e., Co n C- = 0). Hence,

A c B0 A BO c CO = A c C A (Co n C- = 0) = A0 n Co = 0. On the other hand, for

the configuration A o B A B o C, we can not say certainly whether the AO n CO is empty

or nonempty which means A0 n Co = unknown. We can prove this statement by the two









scenarios described in Figure 3-4 where for this same configuration, we get different

interior-interior intersection values between A and C.









(a) (b)

Figure 3-4. The interiors of A and C: (a) intersects, (b) does not intersect.


Based on the above observations, for each configuration we can determine the

values (i.e., either true or false or unknown) of all 9-intersection predicates between A

and C. Since, we don't need to determine the exterior-exterior intersection because

it is always true. Hence, we define remaining eight of the 9-intersection predicates by

three sets of rules that specify for which configuration, the predicate is supposed to yield

certainly true, certainly false, and unknown. Then by applying some simple propositional

logic reduction techniques and set theory notations (e.g., by combining c and = to

c), the sets of inference rules for each 9-intersection predicates, indexed as P, where

1 < i < 9, are as follows:
true A = B A Bo = Co V

AOB0 A BO c Co V

BOC A Bo c AO V

-AB A OAOB A BO c Co A -,OBC V
P1 : AC=
-BC A OBOC A BO c A A -OAOB

false Ao c Bo A -BC V

-AOBO A CO C Bo

unknown otherwise














P2 : AoC = (


P3 :AC- =















P4 : OACo=


true (C c BO V C o Bo) A Bo C A V

Co = Bo A BO c Ao A -BA

false CO c Bo A BAo V

( CB V B0 C) A Ao C Bo

unknown otherwise

true Co C Bo A -(A C B) V

(Co c B V B o C) A Ao = BO V

B o Co A (BO c AO v A c Bo) V

-BCo A AoBo V

0BoCo A aOBOC A A~ABo A OAOB V

C c B A ,BBBC A AO c Bo A OAOB V

B c Co A OBOC A B C Ao A -OAOB

false Bo C AO A Co C Bo

unknown otherwise

true (Ao c Bo V A o Bo) A Bo C C V

AO = Bo A BO c C A -OBOC

false Ao c Bo A -BCo V

(-AoB V Bo C Ao) A Co C Bo

unknown otherwise



































P6 : AC- = (


true





false


A = B A B = C V

A = Bo A (Bo c CV Co c Bo v -BoCo) A OBOC v

B = Co A (Bo c Ao V Ao c B v -AB) A OAOB

A c B A OABA (BOc C v -BoC) v

A c Bo A OAB A (BO c Co v -BoCo) A -OBOBC v

C Bo A OBBBCA (B C Ao v -AoB) v

C c Bo A OBOC A (BO C A V -AoB) A -OAOB v


unknown otherwise

true C C Bo A (Ao C Bo) V

(Ao c B V A o Bo v (-AoB A -OAB)) A B = Co V

-BCo A A c BO V

-BCo A OBBBC A (-AoB V BO c AO) A BOAB V

C c B A OBBBC A AO c B A OAOB V

B c Co A OBOC A BO c Ao A OAOB

false Bo C AO A Co C Bo

unknown otherwise










A C B A (CoC Bo) V


(AO c BO V A o B) A Bo = CO V

A0 o Bo A (BO c Co V Co c Bo) V

AoBo A BoCo V

P7 : A-C = < AB A -OAB A -BC A OBOC V

A0 c Bo A -OAB A Co c Bo A OBOC V

BO c Ao A OAOB A B c C A -OBOC

false BO C Ao A Co C Bo

unknown otherwise

true Ao C B A (C C B) V

(Ao C Bo V A0 o Bo V (-AoB A -OAOB)) A Bo = Co V

-AB A Co C BO V

-A"Bo A -OAOB A (-BC V BO c Co) A BBBC V
P8 A-C =
A c Bo A -OAB A Co c Bo A BBBC V

B C Ao A AB A B c C Co A OBBC

false B C AO A Co C Bo

unknown otherwise

P : A- C- = true

The proof of these rules are done by the simple set theory logic and proof by

counter-example and drawing which are shown in the previous paragraph. The proof

of all the rules are not given in this document due to space constraints. However,

the completeness of this set of rules follows from the formulation of the rules. Two

regions must have exactly one of the eight interior-interior set relations for any


true









topological relationships and after composing region A and C must hold one of the 64

configurations. Since, the inference rules are formulated considering each configuration

into account, these rules never miss any scenario for which it cannot determine the

9-intersection predicates. Thus, inference rules are complete by formulation.

3.3 Relationship Identifying Process

We evaluate the 9-intersection predicates (called evaluated predicates) of the

topological relationship to be inferred by applying the inference rules defined in the

previous sub section. these evaluated predicates have slightly different characteristics

than the usual 9-intersection predicates. Because, evaluated predicates may have

unknown value where as usual 9-intersection predicates always have deterministic

values (i.e., either true or false). This slightly different characteristic is obvious. Because,

we know that the inferred relationship can be very specific (i.e., a single relationship)

or a disjunction of relationships. If the inferred relationship is very specific then all

the evaluated predicate values are deterministic. On the other hand, if the inferred

relationship is a disjunction of relationships then at least one of the evaluated predicates

must have unknown value. In fact, the evaluated predicates have deterministic value

only for those predicates which agree for all the relationships in that disjunction. Since

we may have indeterministic value, we need one more step to identify the relationships)

from the evaluated predicates.

A simple brute force approach to find out the inferred relationship is to compare

the evaluated matrix against each of the 33 relationship matrices, predicate by

predicate. The problem is that it takes too many comparisons. Since the exterior-

exterior intersection is always true, we have to compare eight of these evaluated

predicates for each matching which means 33 x 8 = 264 comparisons are required

in the worst case. To reduce the number of comparisons we build a decision tree of

these 33 relationships. Table 3-1 shows all 33 possible relationship matrices [7]. We

recursively divide the relationship space based on a predicate value at each level of the








Table 3-1. 33 possible topological relationships between two complex regions.
Matrix 1 Matrix 2 Matrix 3 Matrix 4 Matrix 5 Matrix 6
O 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0
0 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 0 1 0
1 1 1 1 1 1 01 1 0 1 1 1
Matrix 7 Matrix 8 Matrix 9 Matrix 10 Matrix 11 Matrix 12
1 0 1 0 1 0 0 1 0 0 1
1 0 10 1 1 1 0 0 1 1 0 1
(1 0 I) (1 0 I) (1 1 ) 1 1 1 (0 0 1) (1 0 1)


Matrix 14
1 0 1
1 0 1
1 11
Matrix 20
111
0 01
1 11
Matrix 26
111
0 11
1 11
Matrix 32
111
1 1 1
1 11
1 01


Matrix 15
1 0 1
1 1
1 0

Matrix 21
( 1 1
111
0 10
0 01
Matrix 27
1 1 1
111
1 00
1 11
Matrix 33
111
i 1 1
1 11
111


Matrix 16
1 0 1
1 1


111
010
Matrix 28
0 1 0

Matrix 28
111
1 1 1
1 01
101


Matrix 17 Matrix 18
1 0 1 1 1
1 1 1 1 1
1 0 i/ 1 1 1
Matrix 23 Matrix 24
111 111
0 1 0 0 1 1
F1 1) 1 I1 1
1 1 0 1
Matrix 29 Matrix 30
111 1 11
1 0 1 1 1 0
11 i/ 1 1 0 1


tree until we reach a single relationship. For example, 18 relationships (Matrices 1 to 18
in Table 3-1) have false as their interior-boundary intersection value. Thus, we divide the
relationship space so that the relationships 1 to 18 are on one side and the relationships
19 to 33 are on the other side. Next if we look into the relationships 19 to 33, we find
that the relationships 19 to 26 represented by Matrix 19 to 26 has the value false for the
boundary-interior predicate and that the other relationships have the predicate value
true. Therefore, we again divide the relationship space where relationship 19 to 26 is on
one side and relationships 27 to 33 are on the other side. We continue this process until


Matrix 13
1 0 1
0 1 1
S111
Matrix 19

0 01
0 01
Matrix 25
i 1 1
0 11
1 01
Matrix 31

1 10
111









there is only one relationship in a leaf node. At each level, we divide the relationship

space into half as close as possible to attain minimum average path length from the root

to the leaf nodes. Since, there are 33 relationships a balanced binary tree should have

the height [Ig 331 = 6. Our decision tree also has the height six. Though, this tree is

not unique but this tree has the minimum average path length. The complete decision

tree is shown in Figure 3-5. Each inner node has two entries. The entry inside a node

describes the current relationship space that is being considered, and the entry above

the node denotes the predicate that has to be considered to further divide the current

relationship space.


Figure 3-5. Decision tree of the relationship space for complex regions.


With the help of this tree, we design a recursive algorithm IdentifyRelationship

(Figure 3-6) for identifying the inferred relationship. The inputs of the algorithm are

the decision tree (T) and the 9-intersection matrix (Mg) which is the evaluated matrix.

The output is the inferred relationship (Rt). At each node, starting from the root, the

value of the predicate assigned to that node is retrieved from the evaluated matrix and









algorithm IdentifyRelationship
(1) input: T := (V, E)
(2) M9
(3) output: Rt
(4) begin
(5) Step 1: Start with the root e T
(6) Step 2: At each node check the value of the evaluated predicate.
(7) Step 2a: If the predicate value is 0, then follow the left subtree.
(8) Step 2b: If the value is 1, then follow the right subtree.
(9) Step 2c: If the value is unknown, then follow both subtrees.
(10) Step 3: Repeat Step 2 until the leaf nodes are reached in all branches.
(11) Step 4: If a single leaf is reached
(12) then return the corresponding relationship.
(13) else return the disjunction of all corresponding relationships.
end IdentifyRelationship

Figure 3-6. The algorithm IdentifyRelationship.

checked. Depending on the value, we follow either the left, right, or both subtrees.

This process recursively follows down to the tree until a leaf node is reached. If all

the evaluated predicates have deterministic value (i.e., true or false) then only one

leaf node is reached. Otherwise, if any predicate has an indeterministic value (i.e.,

unknown), more than one leaf node is found. In this case, the inferred relationship is the

disjunction of all the corresponding relationships represented by those leaf nodes. The

maximum height of this decision tree is 6. Which means if all the evaluated predicates

have deterministic values then in the worst case, it would take 6 comparisons instead

of 264 comparisons which is a 97% improvement. Since evaluated predicates can have

indeterministic values, we may end up searching through the whole tree in the worst

case. The required number of comparisons to search through the whole tree is equal

to the number of the inner nodes. The decision tree that we show in Figure 3-5 has 32

inner nodes. Consequently, 32 instead of 264 comparisons are sufficient which is an

improvement of 88%.









CHAPTER 4
GLOBAL INFERENCE

As we have already discussed in subsection 2.1.3 that the well accepted way of

carrying out the global inference is by means of the path consistency algorithms. In this

chapter we discuss the problem related to constraint satisfaction algorithms in terms of

databases with frequent updates. Then, we propose a generalized reasoning algorithm

for complex regions.

4.1 Global Inference in Databases with Frequent Updates

The first problem of this approach is the high complexity. Since the algorithm

generates a complete knowledge base, it is required to run only once at the beginning.

Thus, one can argue that the higher running time can be counted as pre-processing

time and that it is a one time overhead. This argument holds when the database is

static or changes rarely. If the database change is frequent then this running time

becomes a big overhead. For example, if a new object is added to the database then the

algorithm should run again with this new information. The same argument holds if there

is a change in any relationship because that change may cause other relationships to

adjust which means the algorithm should run to propagate those updates. In case of the

deletion of an object, only the object and the emanating relationships from it have to be

deleted. Therefore, the O(n3) overhead is incurred almost every time when there is a

change, and this becomes worse when the database is large (i.e., n is large).

The second problem arises in order to answer a complex query. For example,

there are two regions A and C describing two earthquake affected areas. We want to

know if there is any part of state S which was hit by both earthquakes. The answer

can be obtained by looking at the topological relationship between the intersection of

A and S as well as the intersection of C and S. Let the intersections be denoted by

/I and /2 respectively. Our goal is to find the relationship between these two regions.

For this purpose, we need to add these two regions as two nodes in the BSCN and









run the path-consistency algorithm. The algorithm gives us not only the relationship

between /1 and /2 but also the relationships between /1 and all the other nodes as well

as the relationships between /2 and all the other nodes. But we do not need these

extra relationships. Hence, the whole procedure becomes quite inefficient. Moreover,

/I and /2 are temporary regions only and are thrown out of the BSCN after the query

execution. When those temporary regions are thrown out, the BSCN must revert to

its previous state. This means we need to save the previous state of the BSCN when

any such complex query is being asked. Based on these observations, we can argue

that complete knowledge may not be desirable in some cases and path consistency

algorithms are not designed to handle database changes. Hence, our goal is to develop

a different run time strategy to carry out the global inference.

Three scenarios can arise when a query is made to find out the topological

relationship between two regions: (i) the relationship is already known which means

no reasoning is required, (ii) no relationship is available and there are no intermediate

nodes through which we can infer the relationship, and (iii) no relationship is available

but there are some intermediate nodes through which we can infer the relationship.

In terms of a graph, these three scenarios are equivalent of having (i) a direct edge

between the two nodes, (ii) no path between the two nodes, and (iii) at least one path

between the nodes respectively. The first scenario is straightforward so that we have

only to be concerned about the other two scenarios. It is very important to identify

whether it is possible to infer knowledge between two given regions. The reasoning

procedure is a costly process. If we could anticipate that the inference of new knowledge

between two complex regions is impossible before starting the procedure, it would save

us time and resources. But surprisingly the solution is straightforward. Since the BSCN

is a graph, a simple path finding algorithm that assumes one of the two regions as the

source and the other one as the destination can answer this question. A necessary





















h f2

Figure 4-1. A chain of relationships.


condition for reasoning is that there is a path between the nodes representing the two

regions.

Therefore, the first step is to run a path finding algorithm. A path between two target

nodes through a set of intermediate nodes corresponds to the chaining example that

we described before in the Introduction. Figure 4-1 describes the scenario where A and

E are the target nodes and B, C, and D are the intermediate nodes. The relationships

are Ri(A, B), R2(B, C), R3(C, D) and R4(D, E), and our goal is to infer R(A, E). We can

solve this long chain of relationships by simplifying it into a series of compositions of

relationships involving three nodes. Referring to Figure 4-1, we first compose R1(A, B)

and R2(B, C) to get Rx(A, C). Then we compose Rx(A, C) and R3(C, D) to obtain

Ry(A, D). Finally, by composing Ry(A, D) and R4(D, E), we get R(A, E). In the Al

domain, this process is known as forward chaining.

Intuitively, shortest path algorithms are a good choice for a path finding algorithm

because they can give us the path with the minimum number of intermediate nodes;

this might ensure a lower processing time. However, let us consider a configuration with

two chains (paths). First, we assume that A overlaps B and B overlaps C. Second,

we assume that A disjoint D, D contains E, and E contains C (Figure 4-2). From the

first chain the inferred relationship between A and C is the universal relationship, i.e.,
















Figure 4-2. Multiple chains of relationships.

the disjunction of all possible relationships. But from the second chain the inferred

relationship between A and C is disjoint. Though both results are correct, the second,

longer chain gives us the more specific and thus better answer. A similar example can

be shown where the shorter path gives us a more specific and thus better answer. In

fact, this shows that there is no relation between the length of the path and the more

specific answer. This means that by considering one path, we may not obtain the most

specific answer. Hence, we have to consider all possible paths, and the intersections

of all inferred relationships obtained through these paths should give us the most

specific relationship. The problem is, in the worst case, the number of all simple paths

between two nodes in a graph is n! when the graph is complete. Interestingly, this worst

case scenario is actually good for the reasoning process because we don't need any

inference when the knowledge base is complete. Assuming the BSCN is a sparse

graph, an alternative heuristic solution is to consider k-shortest simple paths instead of

all simple paths. The k-shortest path problem is a generalization of the shortest path

problem and determines k paths, instead of one, in an increasing order of length. The

length is measured as the number of hops from source to destination which means the

edges of the BSCN are of unit weight. The worst case complexity for the k-shortest

simple path algorithm is O(m + n log n + k) [25] where n is the number of nodes and m

is the number of edges. If we choose k = n, then the complexity becomes O(n log n)

for n log n >= m. Hence for a large database, the number of paths to be considered

becomes large (i.e., say k = n = 1000) which is a sufficiently good approximation in a

sparse database.









4.2 An Algorithm for Reasoning between Complex Regions


algorithm ReasoningBetweenComplexRegion
(1) input: G := (V, E)
(2) M
(3) a,3 e V
(4) output: R(a,3)
(5) begin
(6) if A .l not null then
(7) return MI,P
(8) k:= 0
(9) repeat
(10) pa,, := find the next best path from a to / in G
(11) // pa, is a list of nodes e G that starts with a, ends with / and
(12) // includes the intermediate nodes
(13) for i in intermediate nodes E pa,,
(14) S,:=Evaluate the set relations between the interiors of Ma,i, M,,+i
(15) Mg:=Evaluate predicates by applying the inference rules(S,)
(16) Rt(a, i + 1) := IdentifyRelationship(Mg)
(17) endfor
(18) R(a, ) := R(a, ) n Rt(a,~ )
(19) k := k +
(20) until there are no paths from a to / or k = I V
(21) return R(a, 3)
end ReasoningBetweenComplexRegion

Figure 4-3. The algorithm ReasoningBetweenComplexRegion.


So far, we have described the two basic steps of the reasoning process. In this
section, we integrate these steps which give us a generalized conceptual model for
reasoning as well as a complete picture of our work. The algorithm is also the starting
point of the implementation of this conceptual model. We employ the k-shortest simple
path algorithm and assume that k is equal to the number of nodes in the BSCN. The
inputs of the algorithm ReasoningBetweenComplexRegion (Figure 4-3) are the BSCN
G, a matrix M, which stores the existing relationships, and the two complex regions a
and / for which we infer the relationship. The matrix M is indexed by (i,1) which means
the topological relationship between the complex objects i and j is stored in the matrix
entry Mij. The output of the algorithm is the inferred relationship. There is a simple









check (line 7) to find out whether the relationship already exists or not. If the relationship

already exists, we simply return this relationship and no reasoning is required. The

reasoning procedure has two loops. The outer loop (lines 9 to 20) executes k-shortest

path algorithm. Each time when we get a new path (i.e., Pa,,), the inner loop (lines 13 to

17) is executed. This inner loop executes the forward chaining process. In this loop, the

composition of relationships is performed in three steps. First, the set relations between

the interiors of the regions in concern are being evaluated (line 14). Then, the evaluation

of the 9-intersection predicates by means of the inference rules is performed (line 15)

and then the inferred relationship is obtained by passing those evaluated predicates to

the relationship identifying process (line 16). In order to find out the most specific result,

we take the intersection of all inferred relationships which are obtained through different

paths (line 19). The complexity of the inner loop depends on the length of the chain

because applying the inference rules and the relationship identifying process requires a

constant amount of time. In a graph the maximum path length between any node can be

I V1 1. Hence, the time complexity of the inner loop is O(n). Since the complexity of the

outer loop is O(n log n), this gives us the total complexity of O(n2 log n). This complexity

is lower than the original BSCN path-consistency algorithm. But the main advantage is

that we only need to run this algorithm when a query is fired. Therefore, this approach

can save a lot of overhead for large dynamic databases. It also solves the complex

query problem because it only computes the relationship of the target objects without

modifying any other relationships in the database.

4.3 Simulation and Results

The performance of the heuristic depends on the percent of time the heuristic is

able to find the most specific relationship between two regions. Since we consider k

paths, instead of all paths, between two nodes representing the two regions, it possible

that we may miss the path which could give us the most specific relationship. Let

assume, the number of paths in a BSCN between any two nodes is E. If k > E, then











Performance of the Heuristic









0 0 0 0 0 00 0

60 -4n s y sy ( w h p, p ) t t te h c g s us te
C0


specific relationship is p k/E since all the edges have equal weights. We generate
-[:20




10 20 30 40 50 100 150 200 250 300 350 400 500
Sof Nodes


Figure 4-4. Performance of the heuristic for different database sizes.


we can surely say (i.e., with probability, p = 1) that the heuristic gives us the most

specific result. On the other hand, if k < E then the probability of obtaining the most

specific relationship is p = k/E since all the edges have equal weights. We generate

a random graph which represents the BSCN. The number of edges of each node is

power law distributed between 1 and n, where n is the number of nodes in the graph.

The reason is that the edges represents the information available about the nodes.

In reality, we have a lot of information for few regions, reasonable amount information

for many regions and less information about rest of the regions. This phenomenon

is captured by the power law distribution. We run the simulation for different sizes of

databases and observe the performance of the heuristic by varying k. At each run, the

performance is measured by averaging the p for all possible pairs of nodes. The number

of considered paths, k is a constant multiple of the number of nodes, i.e., k = cn to

keep the complexity of the k-shortest path bounded to O(n log n). The figure 4-4 shows

that the performance of the heuristic decreases with the increase of the database size









which is expected. The figure 4-4 also shows that for a fixed database size, performance

increases if we consider more paths, i.e., if we increase the c. For small databases

such as 10 < n < 50, the heuristic is able to find the most specific result more than

90% of time which is considered to be good performance by a heuristic. The heuristic

performs reasonably well (i.e., above 80%) in case of medium sized databases with

50 < n < 300. As the number of nodes grows beyond 300 nodes, the heuristics does not

perform well when c < 10. But we see that significant performance gain can be obtained

by considering more paths (e.g., c = 20). Though, increasing c does not hurt the overall

complexity as long as n >> c but it slows the algorithm by the factor of c2 log c2/ci log c

where c2 > c1. Based on this observation, the value of c can be set by the user based

on the size of the database and requirement of the precision.









CHAPTER 5
CONCLUSIONS AND FUTURE WORKS

From an application point of view, more complex geometric structures than the

simple spatial objects are required to represent real world spatial phenomena. It is

often the situation that if the database is large and complex, the complete knowledge

regarding the participating objects is unavailable. The first contribution of this paper is

the design of a complete set of inference rules through which we can infer topological

relationship between complex regions. The inference rules are formulated in such a

way that it can also be applied to the simple regions. Our second contribution is to

define a overall conceptual framework for reasoning process from the database point of

view which can handle the typical database issues like updating, adding and deleting

information.

A main topic for future work is to implement the framework in spatial databases. We

plan to apply some algorithmic (e.g., dynamic programming) and Artificial Intelligence

(e.g., forward chaining, decision tree) techniques to implement this conceptual reasoning

framework. An important topic for future work is to explore other heuristics for global

inference such as using different weights for the edges. In this document we consider

equal weight for each relationship. But an observation, in case of simple regions,

shows that composing any relationship with the overlap relationship always results in

a disjunction of relationships. Hence, it is less probable that most specific result can

be found if a chain has overlap relationship. We can give higher weight to the edges

representing overlap, so that a chain containing overlap is considered later by the

k-shortest path algorithm. Another important topic for future work is extending the

reasoning model to all combinations of complex objects such as line-line and line-region.









REFERENCES


[1] M. Egenhofer, A. Frank, J. Jackson, A topological data model for spatial databases,
in: 1st Int. Symp. on Large Spatial Databases, 1989, pp. 271-286.

[2] M. Egenhofer, Spatial SQL: A query and presentation language, IEEE Transactions
on Knowledge and Data Engineering 6 (1) (1994) 86-95.

[3] R. G0ting, Geo-relational algebra: a model and query language for geometric
database systems, in: Int. Workshop on Computational Geometry on
Computational Geometry and its Applications, Springer-Verlag New York, Inc.,
1988, pp. 90-96.

[4] Randell, D.A., Z. Cui, A. G. Cohn, A spatial logic based on regions and connection,
in: Proceedings 3rd International Conference on Knowledge Representation and
Reasoning, 1992.

[5] R. G0ting, M. Schneider, Realm-based spatial data types: the ROSE algebra, The
VLDB Journal 4 (2) (1995) 243-286.

[6] M. Schneider, Spatial data types for database systems: finite resolution geometry
for geographic information systems, Springer Verlag, 1997.

[7] M. Schneider, T. Behr, Topological relationships between complex spatial objects,
ACM Transactions on Database Systems 31 (1) (2006) 81.

[8] M. Egenhofer, J. Herring, Categorizing binary topological relations between
regions, lines, and points in geographic databases, Technical Report, Department
of Surveying Engineering, University of Maine, Orono, ME.

[9] A. Tarski, On the calculus of relations, The Journal of Symbolic Logic 6 (3) (1941)
73-89.

[10] R. Maddux, Some algebras and algorithms for reasoning about time and space,
Internal paper, Department of Mathematics, Iowa State University, Ames, Iowa.

[11] Z. Cui, A. Cohn, D. Randell, Qualitative and Topological Relationships in Spatial
Databases, Int. Symposium on Advances in Spatial Databases (1993) 296-315.

[12] M. Egenhofer, Deriving the composition of binary topological relations, Journal of
Visual Languages and Computing 5 (2) (1994) 133-149.

[13] M. Egenhofer, Reasoning about binary topological relations, in: 2nd Int. Symp. on
Advances in Spatial Databases, 1991, pp. 143-160.

[14] M. Vasardani, M. Egenhofer, Single-holed regions: Their relations and inferences,
Geographic Information Science (2008) 337-353.

[15] M. Vasardani, M. Egenhofer, Comparing Relations with a Multi-holed Region,
Spatial Information Theory (2009) 159-176.









[16] A. Abdelmoty, B. EI-Geresy, A general method for spatial reasoning in spatial
databases, in: 4th Int. Conf. on Information and Knowledge Management, 1995, pp.
312-317.

[17] M. Grigni, D. Papadias, C. Papadimitriou, Topological inference, in: 14th Int. Joint
Conference on Artificial Intelligence, 1995, pp. 901-907.

[18] V. Haarslev, R. Moller, SBox: A qualitative spatial reasoner's progress report, in:
11th Int. Workshop on Qualitative Reasoning, 1997, pp. 3-6.

[19] R. Dechter, Constraint networks, Encyclopedia of Artificial Intelligence 1 (1992)
276-285.

[20] A. K. Mackworth, E. C. Freuder, The complexity of some polynomial network
consistency algorithms for constraint satisfaction problems, Artificial Intelligence
25 (1) (1985) 65-74.

[21] A. M. University, A. K. Mackworth, E. C. Freuder, The complexity of constraint
satisfaction revisited, Artificial Intelligence 59 (1993) 57-62.

[22] P. V. Beek, On the minimality and decomposability of constraint networks, in: In
Proc. of the 10th National Conference on Artificial Intelligence, 1992, pp. 447-452.

[23] T. Smith, K. Park, Algebraic approach to spatial reasoning, Int. Journal of
Geographical Information Science 6 (3) (1992) 177-192.

[24] J. Renz, B. Nebel, On the complexity of qualitative spatial reasoning: A maximal
tractable fragment of the region connection calculus, Artificial Intelligence 108 (1)
(1999) 69-123.

[25] D. EPPSTEIN, Finding the k shortest paths, SIAM journal on computing 28 (2)
(1999) 652-673.









BIOGRAPHICAL SKETCH

Arif is from the city of Tangail, Bangladesh. He did his bachelor's in computer

science and engineering from Bangladesh University of Engineering and Technology

(BUET) (2006). He worked with GrameenPhone Ltd, the leading telecom company

in the country, in Dhaka as a System Engineer in the Research and Development

Department for almost two years. He did his master's in computer science from the

Computer and Information Science and Engineering Department at the University of

Florida (2010). His research interests include Spatial/Spatio-Temporal Databases,

Spatial Reasoning, Ad hoc Networking and Social Networking.





PAGE 1

TOPOLOGICALREASONINGBETWEENCOMPLEXREGIONSINDATABASESWITHFREQUENTUPDATESByMDARIFULHASANKHANATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2010

PAGE 2

c2010MdArifulHasanKhan 2

PAGE 3

Tomyparents 3

PAGE 4

ACKNOWLEDGMENTS Firstofall,IthankDr.MarkusSchneiderforhisinvaluableguidanceandencouragement.Withouthisguidancethisthesiswouldnothavebeenpossible.IamalsogratefultomysupervisorycommitteemembersDr.JonathanLiuandDr.AlinDobrasuggestionsandfeedbacks.Iamextremelyfortunateforhavingsuchalovingandcaringparents.Theirwordsandsupporthavebeenthemainmotivatingfactorallthroughmyeducation. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 9 2BACKGROUNDANDRELATEDWORK ...................... 11 2.1Background ................................... 11 2.1.1SpatialObjects ............................. 11 2.1.2SpatialRelationships .......................... 11 2.1.3TopologicalReasoningwithComplexRegions ............ 13 2.2RelatedWork .................................. 15 3LOCALINFERENCE ................................. 17 3.1SetRelationshipsbetweentheInteriors ................... 17 3.2InferenceRules ................................. 19 3.3RelationshipIdentifyingProcess ....................... 25 4GLOBALINFERENCE ................................ 29 4.1GlobalInferenceinDatabaseswithFrequentUpdates ........... 29 4.2AnAlgorithmforReasoningbetweenComplexRegions .......... 33 4.3SimulationandResults ............................ 34 5CONCLUSIONSANDFUTUREWORKS ..................... 37 REFERENCES ....................................... 38 BIOGRAPHICALSKETCH ................................ 40 5

PAGE 6

LISTOFTABLES Table page 2-1NumberofTopologicalPredicatesBetweenTwoComplexSpatialObjects. ... 13 3-133possibletopologicalrelationshipsbetweentwocomplexregions. ...... 26 6

PAGE 7

LISTOFFIGURES Figure page 2-1Examplesofa(a)complexpointobject,(b)acomplexlineobject,and(c)acomplexregionobject. ................................ 12 2-2Eightbasictopologicalrelationshipsbetweentwosimpleregions. ........ 12 3-1StepsofLocalInference. .............................. 17 3-2(a)Acomplexregionwithitsfacesandholes,and(b)itsinterior,boundary,andexterior. ..................................... 18 3-3(a)9-IntersectionMatrix,(b)complexregionsAandBmeet,(c)Rmeet(A,B). 18 3-4TheinteriorsofAandC:(a)intersects,(b)doesnotintersect. .......... 21 3-5Decisiontreeoftherelationshipspaceforcomplexregions. ........... 27 3-6ThealgorithmIdentifyRelationship. ......................... 28 4-1Achainofrelationships. ............................... 31 4-2Multiplechainsofrelationships. ........................... 32 4-3ThealgorithmReasoningBetweenComplexRegion. ................ 33 4-4Performanceoftheheuristicfordifferentdatabasesizes. ............ 35 7

PAGE 8

AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceTOPOLOGICALREASONINGBETWEENCOMPLEXREGIONSINDATABASESWITHFREQUENTUPDATESByMdArifulHasanKhanAugust2010Chair:MarkusSchneiderMajor:ComputerEngineering ReasoningaboutspacehasbeenaconsiderableeldofstudybothinArticialIntelligenceandinspatialinformationtheory.Manyapplicationsbenetfromtheinferenceofnewknowledgeaboutthespatialrelationshipsbetweenspatialobjectsonthebasisofalreadyavailableandexplicitspatialrelationshipknowledgethatwecallspatial(relationship)facts.Hence,thetaskistoderivenewspatialfactsfromknownspatialfacts.Aconsiderableamountofworkhasfocusedonreasoningabouttopologicalrelationships(asaspecialandimportantsubsetofspatialrelationships)betweensimplespatialobjectslikesimpleregions.ThereisacommonconsensusintheGISandspatialdatabasecommunitiesthatsimpleregionsareinsufcienttomodelspatialrealityandthatcomplexregionobjectsareneededthatallowmultiplecomponentsandholes.Modelsfortopologicalrelationshipsbetweencomplexregionshavealreadybeendeveloped.Hence,asthenextlogicalstep,thegoalofthisthesisistodevelopareasoningmodelforthem.Furthernoreasoningmodelconsiderschangesofthespatialfactbasisstoredinadatabaseinbetweenthequeries.Weshowthatconventionalmodelingsuffersperformancedegradationwhenthedatabaseisfrequentlychanging.Ourmodeldoesnotassumeanygeometricrepresentationmodelordatastructurefortheregions.Themodelisalsobackwardcompatiblewhichmeansthatitisalsoapplicabletosimpleregions. 8

PAGE 9

CHAPTER1INTRODUCTION UnderstandingthetopologicalrelationshipsbetweenobjectsinspacehasbecomeamultidisciplinaryresearchissueinvolvingAI,CAD/CAMsystems,cognitivescience,computervision,imagedatabases,linguistics,robotics,GIS,andspatialdatabases.FromaspatialdatabaseandGISpointofview,topologicalrelationshipsarenecessaryaslterconditionsforspatialselectionsandspatialjoinsaswellasforspatialdataretrievalandanalysis.InspatialdatabasesandGIS,wegenerallydealwithalargenumberofspatialobjects.Hence,itisnotuncommonthatwedonothaveallpossiblerelationshipsavailablebetweeneverypairofspatialobjectsallthetime.Thissituationcanariseeitherduetoalackofinformationorsinceitisimpossibletogetalltherelationships.Todealwiththisproblemofalackofcompleteknowledge,weneedaprocessthroughwhichwecaninferthetopologicalrelationshipbetweentwospatialobjectswheretherelationshipdoesnotcurrentlyexistintheknowledgebase.Thisprocessiscalledreasoning.Hence,reasoningabouttopologicalrelationshipisamethodofinferringnewtopologicalrelationships,call2D-spatialfacts,betweentwospatialobjectsusingtheotherexistingspatialfactsintheknowledgebase.Forexample,giventhreeobjectsA,BandC,andgiventwotopologicalrelationshipsRx(A,B)andRy(B,C),reasoninghelpsustondouttherelationshipRzbetweenAandCwhereRzdoesnotexistintheknowledgebase.Thisprocessisalsocalledthecompositionofrelationshipswhichisthemostcommonmethodofreasoning. Sofar,themainfocusoftheavailablereasoningmodelsistodealwithsimpleregions.Butintherealworldweoftenfacethesituationwherearealobjectscannotberepresentedbysimpleregionsalone.Forexample,ItalycontainstheVaticanasahole,andtheGalapagosislanddoesnotconsistofasingleislandbutratherofacollectionofmanyislands.Thesespatialphenomenacannotberepresentedbysimpleregions.Thesecondproblemisthatthecurrentreasoningmodelshardlytakethechangesofspatial 9

PAGE 10

factsintoaccount.Itisnaturalthatoftentheinformationisadded,deletedorupdatedinthedatabases.Soitisimportanttounderstandaswellastoconsidertheeffectofsuchchangeswhiledesigningareasoningmodel. Themaingoalofthisthesisistodevelopareasoningmodelforcomplexregions.Themainchallengeistodealwithalargenumberofpossibletopologicalrelationshipsbetweentwocomplexregionsaswellastodealwithalargenumberofsuchregions.Oursecondgoalistoderiveasetofinferencerulesbywhichtheinferenceofrelationshipsisperformed.Sincethetypeforsimpleregionsisasubsetofthetypeforcomplexregions,itisalsoourgoalthatthereasoningmodelisabletohandlesimpleregionswithoutrequiringanymodication.Finally,weshowtheeffectofthechangesofspatialfactsonthereasoningprocess,andweproposeanalgorithmtohandlethosechanges. Weproposeageneralizedprocesstoinfernewrelationshipsbetweencomplexregionswhichisnotrestrictedbythenumberofregionsaswellaschangesinthedatabase.Theprocesshastwobasicsteps.Intherststep,weperformthereasoningprocessinvolvingthreeregions,callitLocalInference.Inthesecondstep,weextendthislocalinferencetoNregionsandhencecallitGlobalInference. Theremainderofthethesisisorganizedasfollows:Section 2 discussesbackgroundandrelatedworkregardingreasoningmodels.InSections 3 and 4 ,wedescribethelocalandglobalinferencerespectively.Finally,Section 5 drawssomeconclusionsanddiscussesfuturework. 10

PAGE 11

CHAPTER2BACKGROUNDANDRELATEDWORK 2.1Background Inthissectionwediscussaboutthedifferenttypesofspatialobjectsfollowedbythedifferenttypesofrelationshipsbetweenspatialobjects.Then,Weexplainthebasicstepsofthereasoningprocessandthegeneralalgorithmstoimplementthesesteps. 2.1.1SpatialObjects Inthepast,numerousdatamodelshavebeenproposedwiththeaimofformulatingspatialobjectsindatabasesandGIS.Spatialobjectsembeddedin2D-spacecanbeofthreetypes:(i)pointobjects,(ii)lineobjects,and(iii)regionobjects.Pointobjectsare0-dimensionalspatialobjectsandonlyprovidepositions.Lineobjectsare1-dimensionallinearspatialobjectsthathavealength.Regionobjectsare2-dimensionalspatialobjectswithanextent(i.e.,bothheightandwidth).Eachkindofspatialobjectcanbecategorizedaseitherasimplespatialobject[ 1 4 ]oracomplexspatialobject[ 5 7 ].Inthisthesiswemainlyconsidercomplexregionobjects.TheFigure 2-1 showsthedifferenttypesofcomplexspatialobjects.Asimpleregionistopologicallyequivalenttoacloseddisc;itdoesnothaveholes.However,acomplexregionmayhavemultiplecomponents,callfacesandmayhavemultipleholes.Oneimportantaspectisthatforthereasoningprocessthespatialobjectsareonlyneededassymbolicterms;theirgeometriesarenotrequired.Themathematicalbasisforformalizingthespatialobjects(i.e.,bothsimpleandcomplex)ispointsettheoryandpointsettopologywhichassumesthattheplanarspaceiscomprisedofaninnitenumberofpoints. 2.1.2SpatialRelationships Therearethreekindsofspatialrelationships:(i)directionalrelationships,(ii)topologicalrelationshipsand(iii)distancerelationships.Directionalrelationshipsvalidatethecardinaldirectionbetweentwospatialobjects(e.g.,north,southwest).Distancerelationshipsvalidatethequalitativedistancebetweenspatialobjects(e.g.,far,close). 11

PAGE 12

Figure2-1. Examplesofa(a)complexpointobject,(b)acomplexlineobject,and(c)acomplexregionobject. Figure2-2. Eightbasictopologicalrelationshipsbetweentwosimpleregions. Ourfocusisontopologicalrelationshipswhichcharacterizetherelativepositionoftwospatialobjects(e.g.,overlap,meet).Forexample,theeightbasictopologicalrelationshipsbetweentwosimpleregionsareshownintheFigure 2-2 .Animportantapproachforcharacterizingthetopologicalrelationshipsbetweenspatialobjectsisknownas9-intersectionmodel[ 8 ].Byusingthismodel,theauthorsin[ 7 ]haveidentiedthetopologicalrelationshipsbetweenanytwocomplexspatialobjectsirrespectiveoftheirtypes.Thirtythreetopologicalrelationshipshavebeenfoundfortwocomplexregions.ThefollowingTable 2-1 showsthenumbertopologicalrelationshipspossiblebetweeneverycombinationofspatialobjects. 12

PAGE 13

Table2-1. NumberofTopologicalPredicatesBetweenTwoComplexSpatialObjects. ComplexPointComplexLineComplexRegion ComplexPoint 5147 ComplexLine 148243 ComplexRegion 74333 2.1.3TopologicalReasoningwithComplexRegions Asmentioned,reasoningmodelswhichcanonlydealwithsimpleregionsarenotenoughtorepresentrealworldscenarios.Forexample,somebiologistswhoareresearchingontheDarwin'stheory,arelookingforapossibleevolutionarylinkbetweenalandspeciesXandanamphibianspeciesYaroundtheGalapagosislands.ThehuntingareasofthespeciesXandYaretheregionsAandCrespectivelyandtheGalapagosislandsistheregionB.TherelationshipsbetweenthehuntingareasofthespeciesXandYwiththeGalapagosislandsareRx(A,B)andRy(B,C)respectively.Havingtheseinformations,thebiologistmaybeabletogetapossiblelinkbetweenthesetwospeciesbylookingattherelationshipRz(A,C)(i.e.,thetopologicalrelationshipbetweentheirhuntingareas)throughthereasoningprocess.Nowtheregionsinquestionare:theGalapagosislandswhichconsistsofmanyislands(i.e.,complexregion)andthelivingareasofthespeciesmayconnedtooneisland(i.e.,simpleregions)ormayextendtomanyoftheseislands(i.e.,complexregions)ormayhavealakeinsideit(e.g.,regionwithholesi.e.,complexregion).Hence,wecanseefromtheexamplethattheregionsA,BandCcanbeallcomplexregionsoranycombinationofsimpleandcomplexregions.AbovescenariocaneasilybeextendedfromthreeregionstoNregions.Therefore,amoregeneralizedandcomprehensivereasoningmodelisrequired. TherststepofthereasoningprocessisthelocalinferenceinvolvingthreeregionsintheformofRx(A,B)andRy(B,C).Here,RxandRyarethespatialfactsbetweenthecomplexregionsA,BandB,Crespectively.Thegoalistondtherelationship 13

PAGE 14

Rz(A,C).Thislocalinferenceiscarriedoutbyaprocesscalledcompositionofrelation-shipsbymeansofasetofinferencerules.Itisimportanttonotethatthecompositionofrelationshipsdoesnotdependontheextentorthespatialcharacteristicsoftheregionsinconcern.Therefore,thecompositionofrelationshipscanbedenotedasRxRy)Rz.Localinferencealoneisnotenoughforinferringrelationshipsbetweentwocomplexregions.ConsiderthechainR1(A,B),R2(B,C),R3(C,D),R4(D,E)oftopologicalrelationshipsamongtheveregionsA,B,C,D,andE.Inthissituation,localinferencealoneisnotsufcienttoinfertherelationshipbetweenAandE.BecauseanintermediateobjectisrequiredthatisinrelationshiptobothAandE.Inthisexamplescenario,suchanintermediateregiondoesnotexist.Thus,theglobalinferencecomesintoplaywhichmakesuseofthecompositionofrelationshipstoinferrelationshipsbetweenanytworegionsintheknowledgebase. Animportantobservationisthattheglobalinferenceisorthogonaltothelocalinference.Thatis,globalinferencecanemployanyalgorithmtoinferrelationshipsgloballyaslongasthecompositionofrelationshipsisavailable.Unsurprisingly,theglobalinferenceisaconstraintsatisfactionproblem.Aconstraintsatisfactionproblem(CSP)isdenedasatriple(X,D,C),whereXisasetofvariables,Disadomainofvalues,andCisasetofconstraints.Everyconstraintisinturnapair(t,R),wheretisatupleofvariablesandRisarelation.TheCSPcanbeviewedasadirectedgraph,wherethenodesarethevariablesandtheedgesbetweentwovariablesaretherelationsortheconstraints.Thisdirectedgraphisalsocalledconstraintnetwork.Inourcase,therelationsareallbinarytopologicalrelationshipsandthevariablesarespatialobjects(i.e.,regions);wecallthisgraphrepresentationbinaryspatialconstraintnetwork(BSCN).TheclassofalgorithmsforglobalinferencingbyusingBSCNisbasedonapathconsistencyprocedure.Apairofvariablesispathconsistentwithathirdvariableifeachconsistentevaluationofthepaircanbeextendedtotheothervariableinsuchawaythatallbinaryconstraintsaresatised.Formally,thevariablesAandCare 14

PAGE 15

pathconsistentwithBifthereisarelationR1(A,C)thatsatisesthebinaryconstraintbetweenAandCandiftherearetworelationsR2(A,B)andR3(B,C)thatsatisfytheconstraintbetweenAandBandbetweenBandC,respectively.Asimpleobservationtellsusthatpathconsistencycanbeachievedthroughcompositionofrelationships.ThealgorithmappliesthepathconsistencyprocedureoverallcombinationsofnodesintheBSCNuntilnonewrelationshipscanbeinferred.Animportantpointisthat,givenapartiallyobservedknowledgebase,thepathconsistencyalgorithmsderivethecompleteknowledge,i.e.,therelationshipsbetweeneverypairofobjects.Thatis,afterrunningtheglobalinferencealgorithmtheknowledgebasebecomescompleteandittakesO(1)timetondtherelationship(s)betweenanypairofthecomplexregions. 2.2RelatedWork Numerousstudieshavebeendoneontopologicalrelationshipsaswellastopologicalreasoning.Thereasoningprocesstriestoinfertherelationshipswhicharedenedandderivedbytherelationshipmodel.Therefore,reasoningmodelsaredependentontheunderlyingrelationshipsmodels.ResearchersfromdifferentdomainssuchasAI,mathematics,GISanddatabases,havebeencontributingtothiseldofstudy.Theauthorsofthepapers[ 9 11 ]attackthisproblemwiththealgebraiclogicapproaches.Theauthorsofthepaper[ 8 ]denedspatialobjectsbasedontopologicalsettheoryandproposed9-IntersectionModelasawaytocharacterizethespatialobjects.Basedonthetopologicalsettheory,theauthorsproposedthereasoningmodelsaboutsimpleregions[ 1 11 13 ],simpleregionswithholes[ 14 15 ].In[ 16 ]theauthorsproposeareasoningmodeltakingtheconcavityoftheregionsintotheaccountalongwiththeirconvexhulls.Mostofthetimes,theinferredrelationshipbetweenspatialobjectsmaynotbeunique,i.e.,theinferredrelationshipcanbeadisjunctionofseveralbasicrelationships.Basedonthisobservation,theauthorsof[ 17 18 ]proposehierarchicalmodelsfortopologicalreasoning. 15

PAGE 16

Alloftheabovementionedstudiesmainlyfocusedonthelocalinference(i.e.,compositionofrelationshipsinvolvingthreeobjectsbymeansofinferencerules).Itiswellunderstoodthatlocalinferenceisanessentialandbasicstepofthereasoningprocessbutwithoutglobalinferencetheprocessisnotcomplete.Thereasonbehindmorefocusonlocalinferenceisbecauseglobalinferenceisaconstraintsatisfactionproblem(CSP)[ 19 22 ]whichisanextensivelystudiedtopicandisindependentofthelocalinferenceprocess.Theauthorsof[ 10 23 24 ]studiedtheissuesrelatedtoconstraintsatisfactionforspatialobjectssuchasthecomplexityandthetractability.Sofar,thelowestcomplexityO(n3)ofCSPalgorithmsisproposedbytheauthorsin[ 20 22 ].AlloftheseCSPalgorithmsoperateonthestaticknowledgebase.Thatis,givenaBSCN,thealgorithmrunsandabletoinferrelationshipsbetweenanypairofcomplexregions.Butovertime,theexistingfactsmaychangeandtheCSPalgorithmsarenotdesignedtohandlechanges.Tobestofourknowledge,noneofthereasoningmodelsdealwiththechangesofthespatialfactsandourworkismotivatedbythisissue. 16

PAGE 17

CHAPTER3LOCALINFERENCE Localinferencetakestwotopologicalrelationships(Rx(A,B)andRy(B,C)),composethemandinferrelationship(s)RZ(A,C).Since9-intersectionmatrixcanuniquelycharacterizeeachtopologicalrelationship,theinputsofthelocalinferencecanbethetwo9-intersectionmatricesandtheoutputisasetofinferredrelationships.TheFigure 3-1 showsthethreestepsofthelocalinferenceprocess.Atrststep,thecorrespondingsetrelationships(i.e.,subsetrelationships,empty/nonemptyintersections)betweentheinteriorsoftheregionsareevaluatedfromthe9-intersectionmatrices.Thentheinferencerulesarebeingappliedtondoutthe9-intersectionpredicatevaluesbetweenAandC.Atthelaststep,theinferredrelationshipsarebeingidentiedfromthepredicatevalues. Figure3-1. StepsofLocalInference. 3.1SetRelationshipsbetweentheInteriors Accordingtothepointsettopology,eachspatialobjectcanbecharacterizedbythreemutuallyexclusivepointsetsinthetopologicalspaceR2.Thesesetsaretheinterior(Ao),theboundary(@A)andtheexterior(A)]TJ /F1 11.955 Tf 7.09 -4.34 Td[()foranyspatialobjectA(Figure 3-2 b).The9-intersectionmodelusesninepredicatestocheckthenineintersectionsofthesepointsetsprovidedbytwospatialobjectsAandBfornon-emptiness.EachtopologicalrelationshipbetweenanytwospatialobjectsischaracterizedbyauniquecombinationofnineBooleanvalues.The9-intersectionpredicatesarearrangedina9-intersectionmatrix(Figure 3-3 a). 17

PAGE 18

Figure3-2. (a)Acomplexregionwithitsfacesandholes,and(b)itsinterior,boundary,andexterior. Ontheotherhand,theinterior,boundary,andexteriorofaspatialobjectareuniquelydenedanddisjointfromeachother[ 7 ].Therefore,accordingtotheregularizeddenitionofcomplexregions,itissufcienttospecifyanyofthesethreesetstouniquelycharacterizearegionobject.Inthisdocumentweconsidertheinteriorofacomplexregiontouniquelycharacterizeit.Hence,foreachtopologicalrelationships,thereisasetrelationbetweentheinteriorsofthetwocomplexregions.Thatis,eithertheinteriorofAisasubsetorsupersetorequalordisjointoroverlapstheinteriorofB.In[ 8 ]theauthorsshowedthewaytondoutthesetrelationshipbetweenanytwocomponentsofaregionobjectfromthe9-intersectionmatrixbyusingthetopologicalpropertiesofthespatialregions.Weemploythatsametechniquetondoutsetrelationbetweentheinteriorsofthetwoparticipatingregionsofatopologicalrelationship. 0@Ao\Bo6=?Ao\@B6=?Ao\B)]TJ /F2 11.955 Tf 10.41 -4.34 Td[(6=?@A\Bo6=?@A\@B6=?@A\B)]TJ /F2 11.955 Tf 10.4 -4.34 Td[(6=?A)]TJ /F2 11.955 Tf 9.74 -4.33 Td[(\Bo6=?A)]TJ /F2 11.955 Tf 9.74 -4.33 Td[(\@B6=?A)]TJ /F2 11.955 Tf 9.75 -4.33 Td[(\B)]TJ /F2 11.955 Tf 10.4 -4.33 Td[(6=?1A 0@0010111111A(a)(b)(c) Figure3-3. (a)9-IntersectionMatrix,(b)complexregionsAandBmeet,(c)Rmeet(A,B). 18

PAGE 19

3.2InferenceRules Fromsettheory,twonon-emptysetsXandYmusthaveoneofthefollowingverelations:(i)XisapropersubsetofY,(ii)XisequaltoY,(iii)YisapropersubsetofX,(iv)XandYhavesomecommonandsomedifferentelements,and(v)XandYdonothaveanycommonelement.Thefourthrelation,wecallitoverlap,denotesthattwosetshavecommonelementsbutnoneofthemisthepropersubsetoftheother.Weextendtheseverelationstoeightbyaddingspecialcasestotherelations(i),(iii)and(v)usingthespatialproperties.ConsiderXandYastheinteriorsofthetworegionsAandBrespectively.Thentherelation(i)denotesthattheregionAiscompletelyinsidetheregionB.Therecanbetwospecialcasesofthisscenarioandtheyare:(a)AisinsideBandtheirboundariestouchand(B)AisinsideBandtheirboundariesdonottouch.Similarly,thesetwospecialcasesalsoholdforrelation(iii)and(iv). LetAandBdenotestheinteriorsets.Thesymbols,^,:,and=havetheirusualmeaning.Thesymboldenotesthepredicateforoverlap,i.e.,AB,(A\B6=?^A)]TJ /F3 11.955 Tf 12.83 0 Td[(B6=?^B)]TJ /F3 11.955 Tf 12.83 0 Td[(A6=?).Thepredicateforanon-emptyintersection,i.e.,A\B6=?,isdenotedbyAB,andthepredicateforanemptyintersection,i.e.,A\B=?,isdenotedby:AB.So,theeightrelationsbetweentheinteriorsoftworegionobjectsarethefollowing: 1. .AB^:@A@B 2. .AB^@A@B 3. .A=B 4. .AB 5. .:AB^@A@B 6. .:AB^:@A@B 7. .BA^@A@B 8. .BA^:@A@B 19

PAGE 20

Therelations1and2aretwospecialcasesoftheoriginalrelation(i).Similarly,therelations5and6aswellastherelations7and8arespecialcasesoftheoriginalrelations(iii)and(v).Unsurprisingly,thesevebasicandeightextendedrelationscorrespondtotheRCC-5andRCC-8[ 4 24 ]respectively.Mostimportantly,theseeightrelationsholdforanytypeofregionobjects(i.e.,simple,complex).Because,simpleregionisnothingbutasinglecomponentcomplexregionwithoutanyhole.Ontheotherhand,sinceweonlyconsidertheinteriorasawholewhichmeanstheinteriorofacomplexregionistheunionoftheinteriorsofitsallfaces,itdoesnotmatterhowmanyholesandcomponentsareinthatcomplexregion.Sincetheseeightrelationscompletelycharacterizetherelationsbetweentheinteriorsoftwocomplexregions,anyrelationshipbetweentwocomplexregionsAandBmustincludeexactlyoneoftheserelations.Therefore,ifwehaveRx(A,B)andRy(B,C)thenbythetransitivityproperty,theinteriorsofAandCmustbelongtoexactlyoneofthe88=64congurationsoftheserelations.Thatis,foreachrelationbetweenAandB,thereareeightpossiblerelationsbetweenBandCwhichgivesus64congurations. Foreachofthese64congurations,wedeterminethe9-intersectionpredicatevaluesbetweenAandC.Asanexample,forthecongurationAB^:@A@BandBC^:@B@Cbyapplyingsimplesettheorylogicsweget,AB^BC)AC)A\C6=?.ThatmeansforthiscongurationofA,BandB,Ctheinterior-interiorintersectionbetweenA,Cisalwaystrue.Similarly,forthesamecongurationwecanprovethattheinterior-exteriorintersectionbetweenAandCisalwaysfalse.Weknowthatthethreecomponents(i.e.,interior,exteriorandboundary)ofaregionobjectaremutuallyexclusive(i.e.,C\C)]TJ /F5 11.955 Tf 13.4 -4.34 Td[(=?).Hence,AB^BC)AC^(C\C)]TJ /F5 11.955 Tf 10.6 -4.34 Td[(=?))A\C=?.Ontheotherhand,forthecongurationAB^BC,wecannotsaycertainlywhethertheA\CisemptyornonemptywhichmeansA\C=unknown.Wecanprovethisstatementbythetwo 20

PAGE 21

scenariosdescribedinFigure 3-4 whereforthissameconguration,wegetdifferentinterior-interiorintersectionvaluesbetweenAandC. Figure3-4. TheinteriorsofAandC:(a)intersects,(b)doesnotintersect. Basedontheaboveobservations,foreachcongurationwecandeterminethevalues(i.e.,eithertrueorfalseorunknown)ofall9-intersectionpredicatesbetweenAandC.Since,wedon'tneedtodeterminetheexterior-exteriorintersectionbecauseitisalwaystrue.Hence,wedeneremainingeightofthe9-intersectionpredicatesbythreesetsofrulesthatspecifyforwhichconguration,thepredicateissupposedtoyieldcertainlytrue,certainlyfalse,andunknown.Thenbyapplyingsomesimplepropositionallogicreductiontechniquesandsettheorynotations(e.g.,bycombiningand=to),thesetsofinferencerulesforeach9-intersectionpredicates,indexedasPiwhere1i9,areasfollows: P1:AC=8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:trueAo=Bo^Bo=Co_AoBo^BoCo_BoCo^BoAo_:AoBo^@A@B^BC^:@B@C_:BoCo^@B@C^BA^:@A@BfalseAB^:BC_:AB^CBunknownotherwise 21

PAGE 22

P2:A@C=8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:true(CB_CB)^BA_C=B^BA^:@B@AfalseCB^:BA_(:CB_BC)^ABunknownotherwise P3:AC)]TJ /F5 11.955 Tf 10.41 -4.34 Td[(=8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:trueCB^:(AB)_(CB_BC)^A=B_BC^(BA_AB)_:BC^AB_:BC^:@B@C^:AB^@A@B_CB^:@B@C^AB^@A@B_BC^@B@C^BA^:@A@BfalseBoAo^CBunknownotherwise P4:@AC=8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:true(AB_AB)^BC_A=B^BC^:@B@CfalseAB^:BC_(:AB_BA)^CBunknownotherwise 22

PAGE 23

P5:@A@C=8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:trueA=B^B=C_A=B^(BC_CB_:BC)^@B@C_B=C^(BA_AB_:AB)^@A@BfalseAoBo^:@A@B^(BC_:BoCo)_AoBo^@A@B^(BC_:BoCo)^:@B@C_CoBo^:@B@C^(BA_:AoBo)_CoBo^@B@C^(BA_:AoBo)^:@A@B_unknownotherwise P6:@AC)]TJ /F5 11.955 Tf 10.41 -4.34 Td[(=8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:trueCB^:(AB)_(AB_AB_(:AB^:@A@B))^B=C_:BC^AB_:BC^:@B@C^(:AB_BA)^@A@B_CB^:@B@C^AB^@A@B_BC^@B@C^BA^:@A@BfalseBoAo^CBunknownotherwise 23

PAGE 24

P7:A)]TJ /F3 11.955 Tf 7.08 -4.34 Td[(C=8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:trueAB^:(CB)_(AB_AB)^B=C_AB^(BC_CB)_:AB^BC_:AB^:@A@B^:BC^@B@C_AB^:@A@B^CB^@B@C_BA^@A@B^BC^:@B@CfalseBoAo^CBunknownotherwise P8:A)]TJ /F10 11.955 Tf 7.08 -4.34 Td[(@C=8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:trueAB^:(CB)_(AB_AB_(:AB^:@A@B))^B=C_:AB^CB_:AB^:@A@B^(:BC_BC)^@B@C_AB^:@A@B^CB^@B@C_BA^@A@B^BC^:@B@CfalseBoAo^CBunknownotherwise P9:A)]TJ /F3 11.955 Tf 7.08 -4.33 Td[(C)]TJ /F5 11.955 Tf 10.4 -4.33 Td[(=true Theproofoftheserulesaredonebythesimplesettheorylogicandproofbycounter-exampleanddrawingwhichareshowninthepreviousparagraph.Theproofofalltherulesarenotgiveninthisdocumentduetospaceconstraints.However,thecompletenessofthissetofrulesfollowsfromtheformulationoftherules.Tworegionsmusthaveexactlyoneoftheeightinterior-interiorsetrelationsforany 24

PAGE 25

topologicalrelationshipsandaftercomposingregionAandCmustholdoneofthe64congurations.Since,theinferencerulesareformulatedconsideringeachcongurationintoaccount,theserulesnevermissanyscenarioforwhichitcannotdeterminethe9-intersectionpredicates.Thus,inferencerulesarecompletebyformulation. 3.3RelationshipIdentifyingProcess Weevaluatethe9-intersectionpredicates(calledevaluatedpredicates)ofthetopologicalrelationshiptobeinferredbyapplyingtheinferencerulesdenedintheprevioussubsection.theseevaluatedpredicateshaveslightlydifferentcharacteristicsthantheusual9-intersectionpredicates.Because,evaluatedpredicatesmayhaveunknownvaluewhereasusual9-intersectionpredicatesalwayshavedeterministicvalues(i.e.,eithertrueorfalse).Thisslightlydifferentcharacteristicisobvious.Because,weknowthattheinferredrelationshipcanbeveryspecic(i.e.,asinglerelationship)oradisjunctionofrelationships.Iftheinferredrelationshipisveryspecicthenalltheevaluatedpredicatevaluesaredeterministic.Ontheotherhand,iftheinferredrelationshipisadisjunctionofrelationshipsthenatleastoneoftheevaluatedpredicatesmusthaveunknownvalue.Infact,theevaluatedpredicateshavedeterministicvalueonlyforthosepredicateswhichagreeforalltherelationshipsinthatdisjunction.Sincewemayhaveindeterministicvalue,weneedonemoresteptoidentifytherelationship(s)fromtheevaluatedpredicates. Asimplebruteforceapproachtondouttheinferredrelationshipistocomparetheevaluatedmatrixagainsteachofthe33relationshipmatrices,predicatebypredicate.Theproblemisthatittakestoomanycomparisons.Sincetheexterior-exteriorintersectionisalwaystrue,wehavetocompareeightoftheseevaluatedpredicatesforeachmatchingwhichmeans338=264comparisonsarerequiredintheworstcase.Toreducethenumberofcomparisonswebuildadecisiontreeofthese33relationships.Table 3-1 showsall33possiblerelationshipmatrices[ 7 ].Werecursivelydividetherelationshipspacebasedonapredicatevalueateachlevelofthe 25

PAGE 26

Table3-1. 33possibletopologicalrelationshipsbetweentwocomplexregions. Matrix1 0@0010011111A Matrix2 0@0010101111A Matrix3 0@0010111011A Matrix4 0@0010111111A Matrix5 0@1000100011A Matrix6 0@1000101111A Matrix7 0@1001001111A Matrix8 0@1001101011A Matrix9 0@1001101111A Matrix10 0@1010101111A Matrix11 0@1010110011A Matrix12 0@1010111011A Matrix13 0@1010111111A Matrix14 0@1011011111A Matrix15 0@1011101011A Matrix16 0@1011101111A Matrix17 0@1011111011A Matrix18 0@1011111111A Matrix19 0@1110010011A Matrix20 0@1110011111A Matrix21 0@1110100011A Matrix22 0@1110101011A Matrix23 0@1110101111A Matrix24 0@1110110011A Matrix25 0@1110111011A Matrix26 0@1110111111A Matrix27 0@1111001111A Matrix28 0@1111011011A Matrix29 0@1111011111A Matrix30 0@1111101011A Matrix31 0@1111101111A Matrix32 0@1111111011A Matrix33 0@1111111111A treeuntilwereachasinglerelationship.Forexample,18relationships(Matrices1to18inTable 3-1 )havefalseastheirinterior-boundaryintersectionvalue.Thus,wedividetherelationshipspacesothattherelationships1to18areononesideandtherelationships19to33areontheotherside.Nextifwelookintotherelationships19to33,wendthattherelationships19to26representedbyMatrix19to26hasthevaluefalsefortheboundary-interiorpredicateandthattheotherrelationshipshavethepredicatevaluetrue.Therefore,weagaindividetherelationshipspacewhererelationship19to26isononesideandrelationships27to33areontheotherside.Wecontinuethisprocessuntil 26

PAGE 27

thereisonlyonerelationshipinaleafnode.Ateachlevel,wedividetherelationshipspaceintohalfascloseaspossibletoattainminimumaveragepathlengthfromtheroottotheleafnodes.Since,thereare33relationshipsabalancedbinarytreeshouldhavetheheightdlg33e=6.Ourdecisiontreealsohastheheightsix.Though,thistreeisnotuniquebutthistreehastheminimumaveragepathlength.ThecompletedecisiontreeisshowninFigure 3-5 .Eachinnernodehastwoentries.Theentryinsideanodedescribesthecurrentrelationshipspacethatisbeingconsidered,andtheentryabovethenodedenotesthepredicatethathastobeconsideredtofurtherdividethecurrentrelationshipspace. Figure3-5. Decisiontreeoftherelationshipspaceforcomplexregions. Withthehelpofthistree,wedesignarecursivealgorithmIdentifyRelationship(Figure 3-6 )foridentifyingtheinferredrelationship.Theinputsofthealgorithmarethedecisiontree(T)andthe9-intersectionmatrix(M9)whichistheevaluatedmatrix.Theoutputistheinferredrelationship(Rt).Ateachnode,startingfromtheroot,thevalueofthepredicateassignedtothatnodeisretrievedfromtheevaluatedmatrixand 27

PAGE 28

algorithmIdentifyRelationship(1) input:T:=(V,E)(2) M9(3) output:Rt(4) begin(5) Step1:Startwiththeroot2T(6) Step2:Ateachnodecheckthevalueoftheevaluatedpredicate.(7) Step2a:Ifthepredicatevalueis0,thenfollowtheleftsubtree.(8) Step2b:Ifthevalueis1,thenfollowtherightsubtree.(9) Step2c:Ifthevalueisunknown,thenfollowbothsubtrees.(10) Step3:RepeatStep2untiltheleafnodesarereachedinallbranches.(11) Step4:Ifasingleleafisreached(12) thenreturnthecorrespondingrelationship.(13) elsereturnthedisjunctionofallcorrespondingrelationships. endIdentifyRelationship Figure3-6. ThealgorithmIdentifyRelationship. checked.Dependingonthevalue,wefolloweithertheleft,right,orbothsubtrees.Thisprocessrecursivelyfollowsdowntothetreeuntilaleafnodeisreached.Ifalltheevaluatedpredicateshavedeterministicvalue(i.e.,trueorfalse)thenonlyoneleafnodeisreached.Otherwise,ifanypredicatehasanindeterministicvalue(i.e.,unknown),morethanoneleafnodeisfound.Inthiscase,theinferredrelationshipisthedisjunctionofallthecorrespondingrelationshipsrepresentedbythoseleafnodes.Themaximumheightofthisdecisiontreeis6.Whichmeansifalltheevaluatedpredicateshavedeterministicvaluesthenintheworstcase,itwouldtake6comparisonsinsteadof264comparisonswhichisa97%improvement.Sinceevaluatedpredicatescanhaveindeterministicvalues,wemayendupsearchingthroughthewholetreeintheworstcase.Therequirednumberofcomparisonstosearchthroughthewholetreeisequaltothenumberoftheinnernodes.ThedecisiontreethatweshowinFigure 3-5 has32innernodes.Consequently,32insteadof264comparisonsaresufcientwhichisanimprovementof88%. 28

PAGE 29

CHAPTER4GLOBALINFERENCE Aswehavealreadydiscussedinsubsection 2.1.3 thatthewellacceptedwayofcarryingouttheglobalinferenceisbymeansofthepathconsistencyalgorithms.Inthischapterwediscusstheproblemrelatedtoconstraintsatisfactionalgorithmsintermsofdatabaseswithfrequentupdates.Then,weproposeageneralizedreasoningalgorithmforcomplexregions. 4.1GlobalInferenceinDatabaseswithFrequentUpdates Therstproblemofthisapproachisthehighcomplexity.Sincethealgorithmgeneratesacompleteknowledgebase,itisrequiredtorunonlyonceatthebeginning.Thus,onecanarguethatthehigherrunningtimecanbecountedaspre-processingtimeandthatitisaonetimeoverhead.Thisargumentholdswhenthedatabaseisstaticorchangesrarely.Ifthedatabasechangeisfrequentthenthisrunningtimebecomesabigoverhead.Forexample,ifanewobjectisaddedtothedatabasethenthealgorithmshouldrunagainwiththisnewinformation.Thesameargumentholdsifthereisachangeinanyrelationshipbecausethatchangemaycauseotherrelationshipstoadjustwhichmeansthealgorithmshouldruntopropagatethoseupdates.Incaseofthedeletionofanobject,onlytheobjectandtheemanatingrelationshipsfromithavetobedeleted.Therefore,theO(n3)overheadisincurredalmosteverytimewhenthereisachange,andthisbecomesworsewhenthedatabaseislarge(i.e.,nislarge). Thesecondproblemarisesinordertoansweracomplexquery.Forexample,therearetworegionsAandCdescribingtwoearthquakeaffectedareas.WewanttoknowifthereisanypartofstateSwhichwashitbybothearthquakes.TheanswercanbeobtainedbylookingatthetopologicalrelationshipbetweentheintersectionofAandSaswellastheintersectionofCandS.LettheintersectionsbedenotedbyI1andI2respectively.Ourgoalistondtherelationshipbetweenthesetworegions.Forthispurpose,weneedtoaddthesetworegionsastwonodesintheBSCNand 29

PAGE 30

runthepath-consistencyalgorithm.ThealgorithmgivesusnotonlytherelationshipbetweenI1andI2butalsotherelationshipsbetweenI1andalltheothernodesaswellastherelationshipsbetweenI2andalltheothernodes.Butwedonotneedtheseextrarelationships.Hence,thewholeprocedurebecomesquiteinefcient.Moreover,I1andI2aretemporaryregionsonlyandarethrownoutoftheBSCNafterthequeryexecution.Whenthosetemporaryregionsarethrownout,theBSCNmustreverttoitspreviousstate.ThismeansweneedtosavethepreviousstateoftheBSCNwhenanysuchcomplexqueryisbeingasked.Basedontheseobservations,wecanarguethatcompleteknowledgemaynotbedesirableinsomecasesandpathconsistencyalgorithmsarenotdesignedtohandledatabasechanges.Hence,ourgoalistodevelopadifferentruntimestrategytocarryouttheglobalinference. Threescenarioscanarisewhenaqueryismadetondoutthetopologicalrelationshipbetweentworegions:(i)therelationshipisalreadyknownwhichmeansnoreasoningisrequired,(ii)norelationshipisavailableandtherearenointermediatenodesthroughwhichwecaninfertherelationship,and(iii)norelationshipisavailablebuttherearesomeintermediatenodesthroughwhichwecaninfertherelationship.Intermsofagraph,thesethreescenariosareequivalentofhaving(i)adirectedgebetweenthetwonodes,(ii)nopathbetweenthetwonodes,and(iii)atleastonepathbetweenthenodesrespectively.Therstscenarioisstraightforwardsothatwehaveonlytobeconcernedabouttheothertwoscenarios.Itisveryimportanttoidentifywhetheritispossibletoinferknowledgebetweentwogivenregions.Thereasoningprocedureisacostlyprocess.Ifwecouldanticipatethattheinferenceofnewknowledgebetweentwocomplexregionsisimpossiblebeforestartingtheprocedure,itwouldsaveustimeandresources.Butsurprisinglythesolutionisstraightforward.SincetheBSCNisagraph,asimplepathndingalgorithmthatassumesoneofthetworegionsasthesourceandtheotheroneasthedestinationcananswerthisquestion.Anecessary 30

PAGE 31

h Figure4-1. Achainofrelationships. conditionforreasoningisthatthereisapathbetweenthenodesrepresentingthetworegions. Therefore,therststepistorunapathndingalgorithm.ApathbetweentwotargetnodesthroughasetofintermediatenodescorrespondstothechainingexamplethatwedescribedbeforeintheIntroduction.Figure 4-1 describesthescenariowhereAandEarethetargetnodesandB,C,andDaretheintermediatenodes.TherelationshipsareR1(A,B),R2(B,C),R3(C,D)andR4(D,E),andourgoalistoinferR(A,E).Wecansolvethislongchainofrelationshipsbysimplifyingitintoaseriesofcompositionsofrelationshipsinvolvingthreenodes.ReferringtoFigure 4-1 ,werstcomposeR1(A,B)andR2(B,C)togetRx(A,C).ThenwecomposeRx(A,C)andR3(C,D)toobtainRy(A,D).Finally,bycomposingRy(A,D)andR4(D,E),wegetR(A,E).IntheAIdomain,thisprocessisknownasforwardchaining. Intuitively,shortestpathalgorithmsareagoodchoiceforapathndingalgorithmbecausetheycangiveusthepathwiththeminimumnumberofintermediatenodes;thismightensurealowerprocessingtime.However,letusconsideracongurationwithtwochains(paths).First,weassumethatAoverlapsBandBoverlapsC.Second,weassumethatAdisjointD,DcontainsE,andEcontainsC(Figure 4-2 ).FromtherstchaintheinferredrelationshipbetweenAandCistheuniversalrelationship,i.e., 31

PAGE 32

Figure4-2. Multiplechainsofrelationships. thedisjunctionofallpossiblerelationships.ButfromthesecondchaintheinferredrelationshipbetweenAandCisdisjoint.Thoughbothresultsarecorrect,thesecond,longerchaingivesusthemorespecicandthusbetteranswer.Asimilarexamplecanbeshownwheretheshorterpathgivesusamorespecicandthusbetteranswer.Infact,thisshowsthatthereisnorelationbetweenthelengthofthepathandthemorespecicanswer.Thismeansthatbyconsideringonepath,wemaynotobtainthemostspecicanswer.Hence,wehavetoconsiderallpossiblepaths,andtheintersectionsofallinferredrelationshipsobtainedthroughthesepathsshouldgiveusthemostspecicrelationship.Theproblemis,intheworstcase,thenumberofallsimplepathsbetweentwonodesinagraphisn!whenthegraphiscomplete.Interestingly,thisworstcasescenarioisactuallygoodforthereasoningprocessbecausewedon'tneedanyinferencewhentheknowledgebaseiscomplete.AssumingtheBSCNisasparsegraph,analternativeheuristicsolutionistoconsiderk-shortestsimplepathsinsteadofallsimplepaths.Thek-shortestpathproblemisageneralizationoftheshortestpathproblemanddetermineskpaths,insteadofone,inanincreasingorderoflength.ThelengthismeasuredasthenumberofhopsfromsourcetodestinationwhichmeanstheedgesoftheBSCNareofunitweight.Theworstcasecomplexityforthek-shortestsimplepathalgorithmisO(m+nlogn+k)[ 25 ]wherenisthenumberofnodesandmisthenumberofedges.Ifwechoosek=n,thenthecomplexitybecomesO(nlogn)fornlogn>=m.Henceforalargedatabase,thenumberofpathstobeconsideredbecomeslarge(i.e.,sayk=n=1000)whichisasufcientlygoodapproximationinasparsedatabase. 32

PAGE 33

4.2AnAlgorithmforReasoningbetweenComplexRegions algorithmReasoningBetweenComplexRegion(1) input:G:=(V,E)(2) M(3) ,2V(4) output:R(,)(5) begin(6) ifM,notnullthen(7) returnM,(8) k:=0(9) repeat(10) p,:=ndthenextbestpathfromtoinG(11) //p,isalistofnodes2Gthatstartswith,endswithand(12) //includestheintermediatenodes(13) foriinintermediatenodes2p,(14) Si:=EvaluatethesetrelationsbetweentheinteriorsofM,i,Mi,i+1(15) M9:=Evaluatepredicatesbyapplyingtheinferencerules(Si)(16) Rt(,i+1):=IdentifyRelationship(M9)(17) endfor(18) R(,):=R(,)\Rt(,)(19) k:=k+1(20) untiltherearenopathsfromtoork=jVj(21) returnR(,) endReasoningBetweenComplexRegion Figure4-3. ThealgorithmReasoningBetweenComplexRegion. Sofar,wehavedescribedthetwobasicstepsofthereasoningprocess.Inthissection,weintegratethesestepswhichgiveusageneralizedconceptualmodelforreasoningaswellasacompletepictureofourwork.Thealgorithmisalsothestartingpointoftheimplementationofthisconceptualmodel.Weemploythek-shortestsimplepathalgorithmandassumethatkisequaltothenumberofnodesintheBSCN.TheinputsofthealgorithmReasoningBetweenComplexRegion(Figure 4-3 )aretheBSCNG,amatrixM,whichstorestheexistingrelationships,andthetwocomplexregionsandforwhichweinfertherelationship.ThematrixMisindexedby(i,j)whichmeansthetopologicalrelationshipbetweenthecomplexobjectsiandjisstoredinthematrixentryMi,j.Theoutputofthealgorithmistheinferredrelationship.Thereisasimple 33

PAGE 34

check(line7)tondoutwhethertherelationshipalreadyexistsornot.Iftherelationshipalreadyexists,wesimplyreturnthisrelationshipandnoreasoningisrequired.Thereasoningprocedurehastwoloops.Theouterloop(lines9to20)executesk-shortestpathalgorithm.Eachtimewhenwegetanewpath(i.e.,P,),theinnerloop(lines13to17)isexecuted.Thisinnerloopexecutestheforwardchainingprocess.Inthisloop,thecompositionofrelationshipsisperformedinthreesteps.First,thesetrelationsbetweentheinteriorsoftheregionsinconcernarebeingevaluated(line14).Then,theevaluationofthe9-intersectionpredicatesbymeansoftheinferencerulesisperformed(line15)andthentheinferredrelationshipisobtainedbypassingthoseevaluatedpredicatestotherelationshipidentifyingprocess(line16).Inordertondoutthemostspecicresult,wetaketheintersectionofallinferredrelationshipswhichareobtainedthroughdifferentpaths(line19).Thecomplexityoftheinnerloopdependsonthelengthofthechainbecauseapplyingtheinferencerulesandtherelationshipidentifyingprocessrequiresaconstantamountoftime.InagraphthemaximumpathlengthbetweenanynodecanbejVj)]TJ /F5 11.955 Tf 17.22 0 Td[(1.Hence,thetimecomplexityoftheinnerloopisO(n).SincethecomplexityoftheouterloopisO(nlogn),thisgivesusthetotalcomplexityofO(n2logn).ThiscomplexityislowerthantheoriginalBSCNpath-consistencyalgorithm.Butthemainadvantageisthatweonlyneedtorunthisalgorithmwhenaqueryisred.Therefore,thisapproachcansavealotofoverheadforlargedynamicdatabases.Italsosolvesthecomplexqueryproblembecauseitonlycomputestherelationshipofthetargetobjectswithoutmodifyinganyotherrelationshipsinthedatabase. 4.3SimulationandResults Theperformanceoftheheuristicdependsonthepercentoftimetheheuristicisabletondthemostspecicrelationshipbetweentworegions.Sinceweconsiderkpaths,insteadofallpaths,betweentwonodesrepresentingthetworegions,itpossiblethatwemaymissthepathwhichcouldgiveusthemostspecicrelationship.Letassume,thenumberofpathsinaBSCNbetweenanytwonodesisE.IfkE,then 34

PAGE 35

Figure4-4. Performanceoftheheuristicfordifferentdatabasesizes. wecansurelysay(i.e.,withprobability,p=1)thattheheuristicgivesusthemostspecicresult.Ontheotherhand,ifk
PAGE 36

whichisexpected.Thegure 4-4 alsoshowsthatforaxeddatabasesize,performanceincreasesifweconsidermorepaths,i.e.,ifweincreasethec.Forsmalldatabasessuchas10n50,theheuristicisabletondthemostspecicresultmorethan90%oftimewhichisconsideredtobegoodperformancebyaheuristic.Theheuristicperformsreasonablywell(i.e.,above80%)incaseofmediumsizeddatabaseswith50n300.Asthenumberofnodesgrowsbeyond300nodes,theheuristicsdoesnotperformwellwhenc10.Butweseethatsignicantperformancegaincanbeobtainedbyconsideringmorepaths(e.g.,c=20).Though,increasingcdoesnothurttheoverallcomplexityaslongasn>>cbutitslowsthealgorithmbythefactorofc2logc2=c1logc1wherec2>c1.Basedonthisobservation,thevalueofccanbesetbytheuserbasedonthesizeofthedatabaseandrequirementoftheprecision. 36

PAGE 37

CHAPTER5CONCLUSIONSANDFUTUREWORKS Fromanapplicationpointofview,morecomplexgeometricstructuresthanthesimplespatialobjectsarerequiredtorepresentrealworldspatialphenomena.Itisoftenthesituationthatifthedatabaseislargeandcomplex,thecompleteknowledgeregardingtheparticipatingobjectsisunavailable.Therstcontributionofthispaperisthedesignofacompletesetofinferencerulesthroughwhichwecaninfertopologicalrelationshipbetweencomplexregions.Theinferencerulesareformulatedinsuchawaythatitcanalsobeappliedtothesimpleregions.Oursecondcontributionistodeneaoverallconceptualframeworkforreasoningprocessfromthedatabasepointofviewwhichcanhandlethetypicaldatabaseissueslikeupdating,addinganddeletinginformation. Amaintopicforfutureworkistoimplementtheframeworkinspatialdatabases.Weplantoapplysomealgorithmic(e.g.,dynamicprogramming)andArticialIntelligence(e.g.,forwardchaining,decisiontree)techniquestoimplementthisconceptualreasoningframework.Animportanttopicforfutureworkistoexploreotherheuristicsforglobalinferencesuchasusingdifferentweightsfortheedges.Inthisdocumentweconsiderequalweightforeachrelationship.Butanobservation,incaseofsimpleregions,showsthatcomposinganyrelationshipwiththeoverlaprelationshipalwaysresultsinadisjunctionofrelationships.Hence,itislessprobablethatmostspecicresultcanbefoundifachainhasoverlaprelationship.Wecangivehigherweighttotheedgesrepresentingoverlap,sothatachaincontainingoverlapisconsideredlaterbythek-shortestpathalgorithm.Anotherimportanttopicforfutureworkisextendingthereasoningmodeltoallcombinationsofcomplexobjectssuchasline-lineandline-region. 37

PAGE 38

REFERENCES [1] M.Egenhofer,A.Frank,J.Jackson,Atopologicaldatamodelforspatialdatabases,in:1stInt.Symp.onLargeSpatialDatabases,1989,pp.271. [2] M.Egenhofer,SpatialSQL:Aqueryandpresentationlanguage,IEEETransactionsonKnowledgeandDataEngineering6(1)(1994)86. [3] R.Guting,Geo-relationalalgebra:amodelandquerylanguageforgeometricdatabasesystems,in:Int.WorkshoponComputationalGeometryonComputationalGeometryanditsApplications,Springer-VerlagNewYork,Inc.,1988,pp.90. [4] Randell,D.A.,Z.Cui,A.G.Cohn,Aspatiallogicbasedonregionsandconnection,in:Proceedings3rdInternationalConferenceonKnowledgeRepresentationandReasoning,1992. [5] R.Guting,M.Schneider,Realm-basedspatialdatatypes:theROSEalgebra,TheVLDBJournal4(2)(1995)243. [6] M.Schneider,Spatialdatatypesfordatabasesystems:niteresolutiongeometryforgeographicinformationsystems,SpringerVerlag,1997. [7] M.Schneider,T.Behr,Topologicalrelationshipsbetweencomplexspatialobjects,ACMTransactionsonDatabaseSystems31(1)(2006)81. [8] M.Egenhofer,J.Herring,Categorizingbinarytopologicalrelationsbetweenregions,lines,andpointsingeographicdatabases,TechnicalReport,DepartmentofSurveyingEngineering,UniversityofMaine,Orono,ME. [9] A.Tarski,Onthecalculusofrelations,TheJournalofSymbolicLogic6(3)(1941)73. [10] R.Maddux,Somealgebrasandalgorithmsforreasoningabouttimeandspace,Internalpaper,DepartmentofMathematics,IowaStateUniversity,Ames,Iowa. [11] Z.Cui,A.Cohn,D.Randell,QualitativeandTopologicalRelationshipsinSpatialDatabases,Int.SymposiumonAdvancesinSpatialDatabases(1993)296. [12] M.Egenhofer,Derivingthecompositionofbinarytopologicalrelations,JournalofVisualLanguagesandComputing5(2)(1994)133. [13] M.Egenhofer,Reasoningaboutbinarytopologicalrelations,in:2ndInt.Symp.onAdvancesinSpatialDatabases,1991,pp.143. [14] M.Vasardani,M.Egenhofer,Single-holedregions:Theirrelationsandinferences,GeographicInformationScience(2008)337. [15] M.Vasardani,M.Egenhofer,ComparingRelationswithaMulti-holedRegion,SpatialInformationTheory(2009)159. 38

PAGE 39

[16] A.Abdelmoty,B.El-Geresy,Ageneralmethodforspatialreasoninginspatialdatabases,in:4thInt.Conf.onInformationandKnowledgeManagement,1995,pp.312. [17] M.Grigni,D.Papadias,C.Papadimitriou,Topologicalinference,in:14thInt.JointConferenceonArticialIntelligence,1995,pp.901. [18] V.Haarslev,R.Moller,SBox:Aqualitativespatialreasoner'sprogressreport,in:11thInt.WorkshoponQualitativeReasoning,1997,pp.3. [19] R.Dechter,Constraintnetworks,EncyclopediaofArticialIntelligence1(1992)276. [20] A.K.Mackworth,E.C.Freuder,Thecomplexityofsomepolynomialnetworkconsistencyalgorithmsforconstraintsatisfactionproblems,ArticialIntelligence25(1)(1985)65. [21] A.M.University,A.K.Mackworth,E.C.Freuder,Thecomplexityofconstraintsatisfactionrevisited,ArticialIntelligence59(1993)57. [22] P.V.Beek,Ontheminimalityanddecomposabilityofconstraintnetworks,in:InProc.ofthe10thNationalConferenceonArticialIntelligence,1992,pp.447. [23] T.Smith,K.Park,Algebraicapproachtospatialreasoning,Int.JournalofGeographicalInformationScience6(3)(1992)177. [24] J.Renz,B.Nebel,Onthecomplexityofqualitativespatialreasoning:Amaximaltractablefragmentoftheregionconnectioncalculus,ArticialIntelligence108(1)(1999)69. [25] D.EPPSTEIN,Findingthekshortestpaths,SIAMjournaloncomputing28(2)(1999)652. 39

PAGE 40

BIOGRAPHICALSKETCH ArifisfromthecityofTangail,Bangladesh.Hedidhisbachelor'sincomputerscienceandengineeringfromBangladeshUniversityofEngineeringandTechnology(BUET)(2006).HeworkedwithGrameenPhoneLtd,theleadingtelecomcompanyinthecountry,inDhakaasaSystemEngineerintheResearchandDevelopmentDepartmentforalmosttwoyears.Hedidhismaster'sincomputersciencefromtheComputerandInformationScienceandEngineeringDepartmentattheUniversityofFlorida(2010).HisresearchinterestsincludeSpatial/Spatio-TemporalDatabases,SpatialReasoning,AdhocNetworkingandSocialNetworking. 40