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Discrete Geometric Path Analysis in Computational DISCRETE GEOMETRIC PATH ANALYSIS IN COMPUTATIONAL GEOMETRY by Amin Gheibi A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Computer Science CARLETON UNIVERSITY Ottawa, Ontario 2015 © Copyright by Amin Gheibi, 2015 To my parents and my wife ii Abstract The geometric shortest path problem is one of the fundamental problems in Computational Geometry and related fields. In the first part of this thesis, we study the weighted region problem (WRP), which is to compute a geometric shortest path on a weighted partitioning of a plane. Recent results show that WRP is not solvable in any algebraic computation model over rational numbers. Thus, scientists have focused on approximate solutions. We first study the WRP when the input partitioning of space is an arrangement of lines. We provide a technique that makes it possible to apply the existing approximation algorithms for triangulations to arrangements of lines. Then, we formulate two qualitative criteria for weighted short paths. We show how to produce a path that is quantitatively close-to-optimal and qualitatively satisfactory. The results of our experiments carried out on triangular irregular networks (TINs) show that the proposed algorithm could save, on average, 51% in query time and 69% in memory usage, in comparison with the existing method. In the second part of the thesis, we study some variants of the Fr´echet distance. The Fr´echet distance is a well-studied and commonly used measure to capture the similarity of polygonal curves. All of the problems that we studied here can be reduced to a geometric shortest path problem in configuration space. Firstly, we study a robust variant of the Fr´echet distance since the standard Fr´echet distance exhibits a high sensitivity to the presence of outliers. Secondly, we propose a new measure to capture similarity between polygonal curves, called the minimum backward Fr´echetdistance (MBFD). More specifically, for a given threshold ", we are searching for a pair of walks for two entities on the two input polygonal curves such that the union of the portions of required backward movements is minimized and the distance between the two entities, at any time during the walk, is less than or equal to ". Thirdly, we generalize MBFD to capture scenarios when the cost of backtracking on the input polygonal curves is not homogeneous. More specifically, each edge of input polygonal curves has an associated non-negative weight. The cost of backtracking on an edge is the Euclidean length of backward movement on that edge multiplied by the corresponding edge weight. Lastly, for a given graph H, a polygonal curve T , and a threshold ", we propose a geometric algorithm that computes a path, P , in H, and a parameterization of T , that minimize the sum of the length of walks on T and P whereby the distance between the entities moving along P and T is at most ", at any time during the walks. iv Acknowledgements Prima facie, I am grateful to the God for well-being that were necessary to complete this thesis. I wish to express my sincere thanks to Prof. J¨org-R¨udigerSack, my supervisor, for all of his admirable guidance and support. His advice on both research as well as on my career have been priceless. I am also grateful to Prof. Anil Maheshwari, who advised me, through my academic endeavor at Carleton University. The door of his office has always been open to me and his constructive comments have improved the quality of my research significantly. I would like to thank Prof. Carola Wenk for suggesting the map matching topic in this thesis. Also, when I was not able to travel to ACM SIGSPATIAL conference, to present our paper (due to visa issue), she kindly accepted to present our paper. Her comments and suggestions improved this thesis tremendously. I would like to thank Prof. Vida Duj- movi´cfor tremendous comments that have improved my thesis. I am also grateful to Prof. Michiel Smid, for constructive comments and suggestions. He was my first instructor on computational geometry, at Carleton University, who inspired me. I would like to thank Dr. Patrick Boily at School of Mathematics and Statistics, Carleton University, to discuss the statistical analysis. Also, I thank Dr. Andre Pugin, Natural Resources of Canada, Prof. Dariush Motazedian and Prof. Claire Samson, Department of Earth Sciences, Carleton University, for the discussions that lead to the new qualitative measures. I am grateful to Prof. Yusu Wang for communication which contains clarifications to Theorem 4.2 in [124]. I take this opportunity to express gratitude to all of the School faculty members and staff for their help and support. I would like to thank my friends, Dr. Christian Scheffer, Dr. Hamid Zarrabi-zadeh, Dr. Kaveh Shahbaz and Dr. Masoud Omran who helped me a lot with constructive discussions. I also place on record, my sense of gratitude to one and all, who directly or indirectly, have lent their hand in this venture. A special thanks to my parents, brothers, and family. Words cannot express how grateful I am to my mother and father for all of the sacrifices that they have made. At last, but not least, I would like to express appreciation to my beloved wife who made my cold nights in Ottawa, warm and delightful. v Table of Contents Abstract iv Acknowledgements v Chapter 1 Introduction 1 1.1 Introduction and Motivation . .1 1.1.1 Geometric Shortest Path Problem . .3 1.1.2 Fr´echet Distance . .7 1.2 Thesis Outline and Contributions . 10 1.3 Shortest Path Literature Review . 15 1.3.1 Shortest Path in Polygons . 15 1.3.2 Minimum Link Path . 20 1.3.3 Manhattan Shortest Path . 21 1.3.4 Weighted Region Problem (WRP) . 22 1.3.5 Shortest Path in 3D . 25 1.4 Fr´echet Distance Literature Review . 27 1.4.1 Hausdorff Distance . 27 1.4.2 Fr´echet Distance . 28 1.4.3 Coupling Distance . 31 1.4.4 Lower Bound . 32 Chapter 2 Weighted Region Problem in Arrangement of Lines 34 2.1 Preliminaries . 36 2.2 Geometric Properties . 37 vi 2.3 The Construction Algorithm . 44 2.4 Minimality of the SP-Hull . 48 2.5 Conclusion . 49 Chapter 3 Path Refinement in Weighted Regions 50 3.1 Introduction . 50 3.2 Preliminaries and Definitions . 54 3.3 Algorithms . 56 3.3.1 Refinement Algorithm for Triangulations . 57 3.3.2 Refinement Algorithm for Parallel Lines . 71 3.3.3 Refinement Algorithm for Arrangements of Lines . 73 3.4 Experimental Results . 74 3.4.1 Motivation . 74 3.4.2 Experimental Setup . 75 3.4.3 Results . 79 3.4.4 Conclusions of Experiments . 93 3.5 Conclusions . 94 Chapter 4 Similarity of Polygonal Curves in the Presence of Outliers 95 4.1 Introduction . 95 4.1.1 Preliminaries . 97 4.1.2 Problem Definition . 99 4.1.3 Counterexample . 100 4.1.4 New Results . 101 4.2 An Approximation Algorithm . 102 4.3 Improvement . 112 vii 4.3.1 An Auxiliary Lemma ..............................112 4.3.2 Construction of G∗ ...............................115 4.3.3 Improved Algorithms for the MinEx and MaxIn Problems........116 4.3.4 IsFPTASAchievable?.............................118 4.4 Conclusion.........................................121 Chapter 5 Minimum Backward Fr´echet Distance 123 5.1 Introduction........................................123 5.2 Problem Definition . .................................124 5.3 Algorithm.........................................125 5.4 Improvement.......................................129 5.5 Conclusion.........................................132 Chapter 6 Weighted Minimum Backward Fr´echet Distance 134 6.1 Introduction........................................134 6.2 Preliminaries and Problem Definition . ......................136 6.3 Algorithm.........................................137 6.4 Improvement.......................................148 6.4.1 FirstStep.....................................148 6.4.2 SecondStep...................................152 6.5 Conclusion.........................................155 Chapter 7 Minimizing Walking Length in Map Matching 156 7.1 Introduction........................................156 7.2 Preliminaries and Definitions ..............................159 7.3 Algorithm.........................................162 viii 7.4 Improvement . 170 7.5 Weighted Non-planar Graphs . 174 7.6 Conclusion . 174 Chapter 8 Open Problems and Future Work 176 Bibliography 181 ix List of Tables Table 3.1 14 Triangular Irregular Networks (TINs) for experiments . 76 Table 3.2 Comparing refinement process and enhanced sleeve methods . 78 Table 3.3 The result of the methods on five TINs. Number of Edges of the Graph (SES), Pre-processing Time (Tp), Average Query Time (QTav), Accuracy (AC), Average Memory Usage (Mav), Top 5 percent Average Memory Usage (TMav), Method 1: Refinement Process, Method 2: Enhanced Sleeve, Method 3: Hybrid . 81 Table 3.4 The result of the methods on other TINs. Number of Edges of the Graph (SES), Pre-processing Time (Tp), Average Query Time (QTav), Accuracy (AC), Average Memory Usage (Mav), Top 5 percent Average Memory Usage (TMav), Method 1: Refinement Process, Method 2: Enhanced Sleeve, Method 3: Hybrid . 83 Table 3.5 The result of the methods on the Everest TIN with weights rather than slopes. In Random Everest the weights are assigned randomly. In Flat Everest all the weights are the same. Number of Edges of the Graph (SES), Pre-processing Time (Tp), Average Query Time (QTav), Accuracy (AC), Average Memory Usage (Mav), Top 5 percent Average Memory Usage (TMav), Method 1: Refinement Process, Method 2: Enhanced Sleeve, Method 3: Hybrid . 84 Table 3.6 The result of fitting 2-parameter Weibull distribution that shape param- eter k is less than 1. Also the result of Kolmogorov-Smirnov goodness- of-fit test (K-S) is reported. 91 x Table 3.7 Correlation between the measures of the distributions of the TINs and the accuracy of our algorithm on the TINs. 93 xi List of Figures Figure 1.1 a) A simply connected region.
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