Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1995 Parallel Algorithms for Constructing Convex Hulls. Jigang Liu Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Liu, Jigang, "Parallel Algorithms for Constructing Convex Hulls." (1995). LSU Historical Dissertations and Theses. 6029. https://digitalcommons.lsu.edu/gradschool_disstheses/6029 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. 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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. A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 PARALLEL ALGORITHMS FOR CONSTRUCTING CONVEX HULLS 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 Depar tment of Computer Science by Jigang Liu B.S., Beijing University of Aeronautics and Astronautics, 1982 August, 1995 UMI Number: 9609103 UMI Microform 9609103 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Acknowledgements I would like to express my heartfelt appreciation to Professor Doris L. Carver for her invaluable encouragement, support, guidance, and insight throughout the course of my studies and research at Louisiana' State University. I would also like to express my gratitude to my advisory committee members, Professors S. Sitharama Iyengar, Donald H. Kraft, J. Bush Jones, David C. Blouin and Jimmie D. Lawson for their persistent interest and constructive advice. I am very grateful to Professor S. Q. Zheng for his generous advice and support of this research. I am indebted to my parents, Mr. Yi Liu and Mrs. Jing Zhang, and my parent-in- laws, Mr. Yisan Wu and Mrs. Baicong Liu for then- devoted support, endless love and selfless sacrifice. I, especially, would like to extend my sincere gratitude and constant love to my wife, Xiaofei Wu, and daughters, Shuang and Yang. Without their love, support, sacri­ fice, and understanding, I would not have been able to complete this work. Table of Contents Acknowledgements .............................................................................................................. ii List o f T ab les ......................................................................................................................... v List o f Figures ....................................................................................................................... vi Abstract .................................................................................................................................. vii 1. Introduction ....................................................................................................................... 1 1.1. Research in Computational Geometry ................................................................ 2 1.2. Conventional Convex Hull Algorithms ............................................................... 3 1.3. Parallel Convex Hull Algorithms ......................................................................... 7 1.4. Oudine of This Dissertation .................................................................................. 11 2. A New Sequential Convex Hull Algorithm ................................................................ 15 2.1. Introduction ............................................................................................................... 15 2.2. Previous Algorithms ................................................................................................ 16 2.3. A Simplified Algorithm ......................................................................................... 20 2.4. Analysis and Comparison ..................................................................................... 24 2.5. Summary ................................................................................................................... 25 3. An Odd-Even Parallel Convex Hull Algorithm ........................................................ 26 3.1. The Previous Parallel Convex Hull Algorithms on Linear Array ................ 26 3.2. A New Approach for Designing Linear Array Convex Hull A lgorithm .............................................................................................................. 30 3.3. An Odd-Even Parallel Convex Hull Algorithm ............................................... 32 3.4. An Example of the Odd-Even Convex Hull Algorithm .................................. 37 3.5. Summary ................................................................................................................... 40 4. Generalized Odd-Even Convex Hull Algorithms .................................................... 42 4.1. The Mesh-Array Architecture and Previous Algorithms ............................... 42 4.2. A Parallel Convex Hull Algorithm for the Case Where n < p ...................... 43 4.3. A Parallel Convex Hull Algorithm for the Case Where n > p ...................... 53 4.4. A New Mesh-array Convex Hull Algorithm ..................................................... 58 4.5. Summary ................................................................................................................... 59 5. Algorithm Performance Analysis................................................................................. 61 5.1. Maspar Machine....................................................................................................... 61 5.2. PCHSS Implementation System ................................... 64 iii 5.2.1. Interface System of PCHSS ....................................................................... 65 5.2.2. Communication System of PCHSS .......................................................... 67 5.2.3. Parallel Computing System of PCHSS ................................................... 68 5.3. Analysis and Discussion ........................................................................................ 69 5.3.1. Testing Strategies and Testing Cases ....................................................... 70 5.3.2. Speed Up Analysis....................................................................................... 72 5.3.3. Parallel Performance Analysis.................................................................. 76 5.4. Summary ................................................................................................................... 83 6. Conclusion ..................................................................................................................... 86 6.1. Summary of the Dissertation ................................................................................. 86 6.2. Significance of the Research ............................................................................... 89 6.3. Future Research ........................................................................................................ 93 Bibliography......................................................... 95 V ita .......................................................................................................................................... 100 iv List of Tables 1.1 A summary of the sequential convex hull algorithms ........................................ 8 1.2 A summary of the parallel convex hull algorithms ............................................ 12 5.1 Time cost on randomly generated data ................................................................. 74 5.2 Time cost on various sizes of designed d ata ........................................................ 75 5.3 A set of 100 random planar points with 30 runs ................................................. 79 5.4 Test result for