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UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations Title Computing Volumes and Convex Hulls: Variations and Extensions Permalink https://escholarship.org/uc/item/67j3p0zv Author Yildiz, Hakan Publication Date 2014 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Santa Barbara Computing Volumes and Convex Hulls: Variations and Extensions A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science by Hakan Yıldız Committee in Charge: Professor Subhash Suri, Chair Professor Amr El Abbadi Professor John Gilbert June 2014 The dissertation of Hakan Yıldız is approved: Professor Amr El Abbadi Professor John Gilbert Professor Subhash Suri, Committee Chairperson May 2014 Computing Volumes and Convex Hulls: Variations and Extensions Copyright c 2014 by Hakan Yıldız iii Acknowledgements I have no doubt that I would not be able to write this dissertation without the encour- agement and contributions of some wonderful and bright people. I, therefore, would like to devote this section to acknowledge those whose support made the completion of this dissertation possible. First and foremost, I would like to thank my supervisor, Prof. Subhash Suri, for his guidance during my Ph.D. studies. In addition to his continuous encouragement for research and scientific contributions, I think I learned a lot from him in terms of what teaching is about. It is my hope that, if I ever end up in a teaching position in the future, I can put into use what I have learnt from him. Besides my supervisor, I would like to thank the rest of my Ph.D. committee; Prof. Amr El Abbadi and Prof. John Gilbert; for their encouragement, insightful comments and sincere suggestions. This section cannot be complete without acknowledging my labmates: Luca, Pegah, Kyle, Kevin, Lucas, and Jonathan. I am grateful that they generously shared their knowledge with me and made my research experience enjoyable. I also would like to thank my collaborators from outside UCSB; John Hershberger, Pankaj Agarwal, Wuzhou Zhang, and Sariel Har-Peled; for their important contribu- tions to my research. iv Finally, I would like to express my gratefulness to my family. With their uncondi- tional support throughout my entire life, they made it possible for me to become who I am now. v Curriculum Vitæ Hakan Yıldız Education 2014 (Expected) Doctor of Philosophy in Computer Science, University of California, Santa Barbara. 2013 Master of Science in Computer Science, University of California, Santa Barbara. 2008 Bachelor of Science in Computer Engineering, Middle East Technical University, Ankara, Turkey Professional Experience 2010-2013 Research Assistant, Department of Computer Science, UC Santa Bar- bara, CA, USA. 2013 SDE Intern, Microsoft Corporation, Redmond, WA, USA. 2011 SDE Intern, Microsoft Corporation, Redmond, WA, USA. 2008-2010 Teaching Assistant, Department of Computer Science, UC Santa Bar- bara, CA, USA. 2006 Intern, METU Research and Development Center, Ankara, Turkey. 2005 Intern, IBM Turk,¨ Ankara, Turkey. vi Publications P. K. Agarwal, S. Har-Peled, S. Suri, H. Yıldız, W.Zhang Convex Hulls under Uncer- • tainty, Under preparation, 2014 S. Eriksson-Bique, J. Hershberger, V. Polishchuk, B. Speckmann, S. Suri, T. Talvitie, • K, Verbeek, H. Yıldız, Geometric kth Shortest Paths, Under preparation, 2014 S. Suri, K. Verbeek, H. Yıldız, On the Most Likely Convex Hull of Uncertain Points, 21st • European Symposium on Algorithms (ESA), 2013 J. Hershberger, S. Suri, H. Yıldız, A Near-Optimal Algorithm for Shortest Paths Among • Curved Obstacles in the Plane, 29th Symposium on Computational Geometry (SoCG), 2013 H. Yıldız, S. Suri, Computing Klee’s Measure of Grounded Boxes, Algorithmica Online, • 2013 H. Yıldız, S. Suri, On Klee’s Measure for Grounded Boxes, 28th Symposium on Compu- • tational Geometry (SoCG), 2012 H. Yıldız, C. Kruegel, Detecting Social Cliques for Automated Privacy Control in Online • Social Networks, 4th Workshop on Security and Social Networking (SESOC), 2012 H. Yıldız, L. Foschini, J. Hershberger, S. Suri, The Union of Probabilistic Boxes: Main- • taining the Volume, 19th European Symposium on Algorithms (ESA), 2011 H. Yıldız, J. Hershberger, S. Suri, A Discrete and Dynamic Version of Klee’s Measure • Problem, 23rd Canadian Conference on Computational Geometry (CCCG), 2011 vii Achievements and Awards Dean’s fellowship, Univerity of California, Santa Barbara, 2013-2014 • 4th Place in DEFCON CTF hacking competition as a member of Shellphish team, Las • Vegas, USA, 2009 5th Place (among 71 teams) at ACM-Programming Contest Southern California Region- • als 2008 as a member of the team representing University of California Santa Barbara Top Graduate in Middle East Technical University, 2008 • Best Senior Project in METU Computer Engineering Senior Project Competition 2008 • as a member of Opus 144 First Place at METU National Programming Contests 2005-2008 • Gokc¸e¨ Karatas¸2005 Education Award for his accomplisments in the field of Informatics • Silver Medal at 16th International Olympiad in Informatics, Athens, Greece, 11-18 Septem- • ber 2004 Silver Medal at 12th Balkan Olympiad in Informatics, Plovdiv, Bulgaria, 3-9 July 2004 • Gold Medal at 11th National Olympiad in Informatics, Ankara, Turkey, December 2003 • Bronze Medal at 15th International Olympiad in Informatics, Wisconsin, USA, 16-23 • August 2003 Bronze Medal at 11th Balkan Olympiad in Informatics, Iai, Romania, 14-20 July 2003 • Silver Medal at 10th National Olympiad in Informatics, Ankara, Turkey, December 2002 • 5th Place in 2001 Turkish National High School Exam among 560.000 students • viii Fields of Study Major Field: Computational Geometry Studies in Computational Geometry with Professor Subhash Suri ix Abstract Computing Volumes and Convex Hulls: Variations and Extensions Hakan Yıldız Geometric techniques are frequently utilized to analyze and reason about multi- dimensional data. When confronted with large quantities of such data, simplifying ge- ometric statistics or summaries are often a necessary first step. In this thesis, we make contributions to two such fundamental concepts of computational geometry: Klee’s Measure and Convex Hulls. The former is concerned with computing the total vol- ume occupied by a set of overlapping rectangular boxes in d-dimensional space, while the latter is concerned with identifying extreme vertices in a multi-dimensional set of points. Both problems are frequently used to analyze optimal solutions to multi- objective optimization problems: a variant of Klee’s problem called the Hypervolume Indicator gives a quantitative measure for the quality of a discrete Pareto Optimal set, while the Convex Hull represents the subset of solutions that are optimal with respect to at least one linear optimization function. In the first part of the thesis, we investigate several practical and natural variations of Klee’s Measure Problem. We develop a specialized algorithm for a specific case of Klee’s problem called the “grounded” case, which also solves the Hypervolume Indi- cator problem faster than any earlier solution for certain dimensions. Next, we extend x Klee’s problem to an uncertainty setting where the existence of the input boxes are defined probabilistically, and study computing the expectation of the volume. Addi- tionally, we develop efficient algorithms for a discrete version of the problem, where the volume of a box is redefined to be the cardinality of its overlap with a given point set. The second part of the thesis investigates the convex hull problem on uncertain input. To this extent, we examine two probabilistic uncertainty models for point sets. The first model incorporates uncertainty in the existence of the input points. The second model extends the first one by incorporating locational uncertainty. For both models, we study the problem of computing the probability that a given point is contained in the convex hull of the uncertain points. We also consider the problem of finding the most likely convex hull, i.e., the mode of the convex hull random variable. xi Table of Contents Acknowledgements iv Curriculum Vitæ vi Abstract x Table of Contents xii List of Figures xv Introduction 1 1 Computing Klee’s Measure of Grounded Boxes 12 1.1 Introduction :::::::::::::::::::::::::::::: 12 1.1.1 Klee’s Measure Problem :::::::::::::::::::: 13 1.1.2 Problem and Results :::::::::::::::::::::: 15 1.2 Dimension Reduction: Sweeping and Weighting ::::::::::: 18 1.3 Weighted Volume of Halfspaces: the Planar Case ::::::::::: 23 1.3.1 Vertical and Horizontal Gradients ::::::::::::::: 24 1.3.2 Intersecting Gradient Volumes via Partial Sums :::::::: 27 1.4 Multidimensional Weighted Volume :::::::::::::::::: 29 1.4.1 Intersecting the Gradients ::::::::::::::::::: 31 1.4.2 Maintaining the Sum of Ordered Products ::::::::::: 34 1.4.3 Higher Order Partial Sums and Sum of Ordered Products ::: 38 1.5 Dynamic Weighted Volume For Arbitrary Boxes ::::::::::: 40 1.5.1 The Planar Case :::::::::::::::::::::::: 41 1.5.2 The d-dimensional Case :::::::::::::::::::: 48 1.5.3 Weighted Volume of Axis-parallel Strips ::::::::::: 51 xii 1.6 Klee’s Measure for d-Grounded Boxes ::::::::::::::::: 54 1.7 Conclusion ::::::::::::::::::::::::::::::: 55 2 The Union of Uncertain Boxes 57 2.1 Introduction :::::::::::::::::::::::::::::: 57 2.2 Probabilistic Volume: Expectation and Tail Bounds :::::::::: 60 2.3 Maintaining the Expected Measure
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