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Parallel Geometric Algorithms. Fenglien Lee Louisiana State University and Agricultural & Mechanical College Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1992 Parallel Geometric Algorithms. Fenglien Lee Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Lee, Fenglien, "Parallel Geometric Algorithms." (1992). LSU Historical Dissertations and Theses. 5391. https://digitalcommons.lsu.edu/gradschool_disstheses/5391 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Order Number 9302909 Parallel geometric algorithms Lee, Fenglien, Ph.D. The Louisiana State University and Agricultural and Mechanical Col., 1992 UMI 300 N. ZeebRd. Ann Arbor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PARALLEL GEOMETRIC ALGORITHMS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Computer Science *>y Fenglien Lee B.Ed., National Taiwan Normal University, 1980 M.S., Indiana University - Bloomington, 1984 August 1992 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In memory of my beloved father Teh-Ming Lee (Oct 22,1911 - Aug. 6,1991) ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I express ray heartfelt appreciation to my advisor, Professor Si-Qing Zheng, for his help, advice, guidance and encouragement throughout the course of my study and research at LSU. I greatly appreciate my advisory committee members, Prof. S. Sitharama Iyengar, Prof. J. Bush Jones, Prof. John M. Tyler, Prof. Charles A. Harlow, and Prof. James G. Oxley, for their precious advices and suggestions about my research. I also express my gratitude to Prof. Peter P. Chen, Prof. Donald L. Greenwell, and Prof. Jianhua Chen for their guidance and help during the period of this work. I would like to thank my good friends Jin-Jwei Chen, Hla Min, Maung Maung Htay, Weian Deng, Chih-Ping Chu, Kyung Hee Kwon, Sangbum Lee, Ching-Liang Tseng and Jigang Liu for their help, encouragement and concern during the past years. Special thanks go to my mother, Wei Yang Lee, my wife, Shujen Chen, and my two sons, Chung-Hau Lee and Chih-Hsuen Lee, for their constant love, sacrifice, and support throughout my life. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Acknowledgements ....................................................................................................... iii List of T ab les.................................................................................................................. vi List of F ig u re s................................................................................................................ vii A b stra c t........................................................................................................................... viii Chapter 1 Introduction ............................................................................................. 1 1.1 Basic Geometric Problems................................................................................ 2 1.2 Computation Models for Parallel Geometric Algorithms ............................. 5 1.3 Outline of the Dissertation ................................................................................ 8 Chapter 2 The Maxima Problem ............................................................................ 11 2.1 Introduction ......................................................................................................... 11 2.2 Preliminaries ....................................................................................................... 13 2.2.1 Data Structures............................................................................................ 13 2.2.2 Hypercube Machine and Data Communications ................................... 14 2.3 Parallel Maxima Algorithm .............................................................................. 19 2.3.1 The Main Algorithm ................................................................................... 19 2.3.2 The Subalgorithm FILTER_S ................................................................... 22 2.3.3 The Subalgorithm FILTER_G................................................................... 23 2.3.3.1 Case 1 of FILTER_G ........................................................................... 26 2.3.3.2 Cases 2 to 6 of FILTER_G ................................................................. 30 2.3.3.3 Proof of Correctness of FILTER_G .................................................. 33 2.4 Analysis of Time Complexity ........................................................................... 34 2.5 Generalizations and Concluding Remarks ...................................................... 37 Chapter 3 The Voronoi Diagram ............................................................................ 41 3.1 Introduction ......................................................................................................... 41 3.2 Preliminaries ....................................................................................................... 42 3.2.1 Definition of Planar Voronoi Diagram ..................................................... 42 3.2.2 Mesh of Trees and Data Communications............................................... 44 3.2.3 Constructing Voronoi Diagram by Geometric Inversion ...................... 46 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 Constructing Voronoi Diagram on a Mesh of Trees ...................................... 48 3.3.1 Constructing Voronoi Diagram from Convex H ull ................................. 48 3.3.2 Constructing Three-Dimensional Convex Hull ...................................... 51 3.3.3 Algorithm for Merging Two Convex Hulls.............................................. 52 3.3.3.1 Determining Internal and External Faces ......................................... 54 3.3.3.2 Identifying Critical External Faces ................................................... 57 3.3.3.3 Generating New Faces ........................................................................ 60 3.3.3.4 Load Balancing Operations ................................................................. 64 3.3.3.5 Analysis of Time Complexity ............................................................ 68 3.3.3.6 Generalizations of the Convex Hull Algorithm .............................. 69 3.4 Discussions ............................................................................................................ 70 Chapter 4 The Congruent Patterns ........................................................................ 71 4.1 Introduction ........................................................................................................... 71 4.2 Finding Congruent Patterns on a Mesh of Trees ............................................. 73 4.2.1 Preprocessing ............................................................................................... 75 4.2.2 Checking Edge-Length Equality of P and G ............................................ 79 4.2.3 Finding Single Congruent Pattern
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