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Parallel Techniques for Paths, Visibility, and Related Problems (Thesis) Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1992 Parallel Techniques for Paths, Visibility, and Related Problems (Thesis) Ziyi Danny Chen Report Number: 92-051 Chen, Ziyi Danny, "Parallel Techniques for Paths, Visibility, and Related Problems (Thesis)" (1992). Department of Computer Science Technical Reports. Paper 972. https://docs.lib.purdue.edu/cstech/972 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. PARALLEL TECHNIQUES FOR PATHS, VISIBILITY, AND RELATED PROBLEMS Ziyi Danny Chen CSD·TR-92·0S1 August 1992 PARALLEL TECHNIQUES FOR PATHS, VISIBILITY, AND RELATED PROBLEMS A Thesis Submitted to the Faculty of Purdue University by Ziyi Danny Chen In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 1992 11 To the memory of my father III ACKNOWLEDGMENTS There are many people that I would like to thank. First, I would like to express my greatest appreciation to my major professor Mikhail J. Atallah. He has always given me the freedom to develop my own research interest and at the same time provided all the support, guidance, and encouragement I needed. Learning how to do research from him and discussing problems with him have become some of my most joyful experience at Purdue. Furthermore, I am greatly indebted to Professor Greg N. Frederickson and Professor Susanne E. Hambrusch, for serving as my committee members and for giving me many needed suggestions and helps during my study at Purdue. I also like to thank Professor Christoph M. Hoffmann for serving as my committee member. I would like to thank the people in the CS department with whom I had discussed research problems, especially Professor Chandrajit Bajaj, Professor Chia-Hoang Lee, Scott McFaddin, Jyh-Jong Tsay, and Tony Tu. I would also like to thank all the people who have been my friends and helped me during my stay at Purdue. In particular, I thank Peter Ng, Thomas Tsang, Rose Ng, and Hiu Leung for many very needed helps during my early stage at Purdue and for many warm dinners and exciting badminton games, my office mates JiaXun Yu and Ching-Shoei Chiang for all the interesting chats and for giving me various helps, my office mate Susan Stratton for tolerating my frequent interruptions and kindly answering all my English questions, and Jindong Chen, Jung-Hong Chuang, Peggy Fahl, D.-J. Guan, Ajay Gupta, Hsin Pan, Bonita Rais, and many Chinese students for their friendship. I would certainly like to thank my "American parents in law," Dick and Gladys, for their love. I enjoy my weekend visits to their house, which always make me feel like being at home. IV I can never thank enough my father and mother. They gave me the most love and support. My father's love for education was my first motivation to pursue a Ph.D. degree. His inspiration, expectation, and confidence have been and will always be one of the greatest sources of support to me. My mother is great; she devoted all her time to raising and educating her eight children. lowe a great deal to my grandfather YueYe, whom I never have a chance to meet. It was his lifetime hard work that provided for my education for years. Many of my thanks go to my sister HuaYing; she and I shared the same dream of getting a higher education. She gave me the best support she could come up with. I thank all my brothers and sisters for their encouragement to my study. I cannot forget it was my uncle GengOn and aunt May who created the opportunity for me to study in the United States and provided a place for me to stay during my first three years in San Francisco. Finally, I am deeply grateful to my dear wife Xiaobo, for her love and patience, for her understanding and appreciation of my work, and for driving many times on the icy and endless road between Detroit and West Lafayette to bring her support and encouragement to me. This research was partially supported by the Office of Naval Research under Grants N00014-84-K-0502 and N00014-86-K-0689, the Air Force Office of Scientific Research under Grant AFOSR-90-0107, the National Science Foundation under Grant DCR­ 8451393, and the National Library of Medicine under Grant R01-LM05118. v TABLE OF CONTENTS Page LIST OF FIGURES Vll ABSTRACT ..... IX 1. INTRODUCTION . 1 1.1 The Parallel Computational Model 3 1.2 Some Basic Parallel Operations and Techniques 6 1.3 Previous Work 9 1.4 Our Thesis. .................... 12 2. RECTILINEAR SHORTEST PATHS WITH RECTANGULAR OBSTACLES 15 2.1 Preliminaries . 18 2.2 Computing a Staircase Separator 26 2.3 Other Building Blocks ...... 30 2.4 Computing the Lengths Matrix D p When IPI = O(IRI) . 34 2.5 Path Lengths between Arbitrary Points . 36 2.5.1 Some Useful Observations ... 36 2.5.2 The B(P)-to-VR Path Lengths .. 44 2.5.3 The VR-to- VR Path Lengths ... 45 2.5.4 Path Lengths with Arbitrary Query Points 51 2.6 Path Lengths When !PI >> IRI ....... 52 2.7 Computing the Actual Paths. ........... 55 2.8 A Note on the Sequential Time Complexity ... 58 3. VISIBILITY OF A SIMPLE POLYGONAL CHAIN FROM A POINT 61 3.1 Preliminaries .... ........... 63 3.2 An Overview of the Algorithm. ..... 66 3.3 Visibility Chains and Their Intersections 68 3.3.1 Simple Geometric Facts ..... 68 3.3.2 Simple Computational Observations. 73 VI Page 3.3.3 The Relative Positions of Ai and VIS(Ci) ..... 74 3.3.4 Computing the Portions of VIS(Ci) Hidden by Ai 78 3.4 The EREW PRAM Implementation. 84 3.5 Applications......................... 86 4. SOLVING GEOMETRIC PROBLEMS ON THE EREW PRAM. 87 4.1 Notation and Basic Algorithm Structure ... 90 4.2 Rank Trees and Parallel Operations ...... 91 4.3 Computing the Convex Hull of Sorted Points. 94 4.4 Triangulating a Trapezoidally Decomposed Polygon 97 4.5 Other Geometric Algorithms . 99 5. DETECTING WEAK VISIBILITY OF A SIMPLE POLYGON 102 5.1 Preliminaries ....... 105 5.2 Some Useful Observations .. ............... 107 5.3 Detecting the Weak Visibility of a Star-Shaped Polygon. 113 5.3.1 Phase 1.A ................ 114 5.3.2 Phase 1.B ...................... 129 5.3.3 Computing the Union of Bad Intervals ...... 131 5.4 Checking the Weak Visibility of a Polygon from an Edge 132 5.4.1 The Preprocess 133 5.4.2 The Basic Idea 134 5.4.3 Phase 2.A ... 138 5.4.4 Phase 2.B ... 144 5.5 Detecting the Weak Visibility of a Simple Polygon. 152 5.5.1 The Reduction by Sack and Suri 152 5.5.2 Our Reduction Procedure .. ........ 154 5.6 Applications...................... 162 5.6.1 Computing Shortest Paths in a Weakly Visible Polygon. 163 5.6.2 Triangulating a Weakly Visible Polygon . 166 5.6.3 Solving the One-Cruising-Guard Problem. 167 5.6.4 Other Applications 170 6. CONCLUSION . 172 BIBLIOGRAPHY .................." 176 VITA 187 VB LIST OF FIGURES 2.1 A rectilinear shortest path between v and w. 16 2.2 Illustrating MAXNE(R') and MAXsw(R'). .. 20 2.3 Illustrating Env(R') and the circular ordering on Bound(Env(R')). 22 2.4 Illustrating B(Q). 23 2.5 Illustrating Monge and non-Monge matrices of path lengths. 24 2.6 Illustrating NE(p) and WS(p). .. 27 2.7 Illustrating the algorithm for Sep. 29 2.8 Illustrating array Vert. 31 2.9 Illustrating Lemma 2.8. 33 2.10 Illustrating the proof of Theorem 2.2. 35 2.11 Illustrating U, U', W, and W' at a node v of T. 37 2.12 Illustrating the proof of Lemma 2.14. 42 2.13 Illustrating Lemma 2.15. ........ 44 2.14 Illustrating the computation of (a) mode 1, and (b) mode 2. 49 2.15 Illustrating the partition of Bound(P). ... 53 3.1 An example of the visibility chain VIS(P). 62 3.2 Illustrating the definitions. ..... 65 3.3 Illustrating the proof of Lemma 3.1. 70 3.4 Illustrating the two possible two-intersection cases. 72 VIll Figure Page 3.5 Illustrating the proofs of Lemmas 3.4 and 3.5. 81 4.1 The convex hull for a set S of planar points. 94 4.2 A triangulation of a monotone polygon. ... 97 5.1 The weakly visible edges of P are ell e2, and e4. 103 5.2 Illustrating HE; and HE;. E.g., HEi = {e3' e4} and HE;; = {es}. 105 5.3 Illustrating the proof of Lemma 5.5. (a) Vj is in Q. (b) Vj is not in Q.. 112 5.4 The four possible cases of IICP(Cj)l. 117 5.5 Illustrating the proof of Lemma 5.6. 119 5.6 Partitioning P into PI' P2 , and P3 • 133 5.7 Illustrating Lemma 5.10. ...... 135 5.8 Illustrating the proof of Lemma 5.12. 137 5.9 Illustrating the proof of Lemma 5.14. 141 5.10 Illustrating the structure of tree Te. 143 5.11 Illustrating Lemma 5.15. ....... 145 5.12 Partitioning Pinto Lp , Mp , and R p . 153 5.13 Illustrating the case where the procedure fails to find p*. 160 5.14 Partitioning P based on edge e. .... 163 5.15 Computing the shortest path SP(p, q). 164 5.16 Illustrating the two types of characteristic intervals. 169 IX ABSTRACT Chen, Ziyi Danny. Ph.D., Purdue University, August 1992.
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