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Computational and Communication Complexity of Geometric Problems

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Computational and Communication Complexity of Geometric Problems Computational and Communication Complexity of Geometric Problems by Sima Hajiaghaei Shanjani B.Sc. Amirkabir University of Technology, 2011 B.Sc. Amirkabir University of Technology, 2012 M.Sc. Sharif University of Technology, 2015 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Computer Science c Sima Hajiaghaei Shanjani, 2021 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. We acknowledge with respect the Lekwungen peoples on whose traditional territory the university stands and the Songhees, Esquimalt and WSÁNEĆ peoples whose ¯ historical relationships with the land continue to this day. ii Computational and Communication Complexity of Geometric Problems by Sima Hajiaghaei Shanjani B.Sc. Amirkabir University of Technology, 2011 B.Sc. Amirkabir University of Technology, 2012 M.Sc. Sharif University of Technology, 2015 Supervisory Committee Dr. Valerie King, Supervisor (Department of Computer Science) Dr. Venkatesh Srinivasan, Departmental Member (Department of Computer Science) Dr. William Evans, Outside Member (Department of Computer Science, University of British Columbia ) iii ABSTRACT In this dissertation, we investigate a number of geometric problems in different settings. We present lower bounds and approximation algorithms for geometric prob- lems in sequential and distributed settings. For the sequential setting, we prove the first hardness of approximation results for the following problems: • Red-Blue Geometric Set Cover is APX-hard when the objects are axis-aligned rectangles. c • Red-Blue Geometric Set Cover cannot be approximated to within 2log1−1/(log log m) m in polynomial time for any constant c < 1/2, unless P = NP , when the given objects are m triangles or convex objects. This shows that Red-Blue Geometric Set Cover is a harder problem than Geometric Set Cover for some class of objects. • Boxes Class Cover is APX-hard. We also define MaxRM-3SAT, a restricted version of Max3SAT, and we prove that this problem is APX-hard. This problem might be interesting in its own right. In the distributed setting, we define a new model, the fixed-link model, where each processor has a position on the plane and processors can communicate to each other if and only if there is an edge between them. We motivate the model and study a number of geometric problems in this model. We prove lower bounds on the communication complexity of the problems in the fixed-link model and present approximation algorithms for them. We prove lower bounds on the number of expected bits required for any random- ized algorithm in the fixed-link model with n nodes to solve the following problems, when the communication is in the asynchronous KT1 model: • Ω(n2/ log n) expected bits of communication are required for solving Diameter, Convex Hull, or Closest Pair, even if the graph has only a linear number of edges. • Ω(min{n2, 1/}) expected bits of communications are required for approximat- ing Diameter within a 1 − factor of optimal, even if the graph is planar. iv • Ω(n2) bits of communications is required for approximating Closest Pair in a c c nc−1/2 graph on an [n ] × [n ] grid, for any constant c > 1 + 1/(2 lg n), within 4 − factor of optimal, even if the graph is planar. We also present approximation algorithms in geometric communication networks with n nodes, when the communication is in the asynchronous CONGEST KT1 model: • An -kernel, and consequently (1 − )–Approximate Diameter and - √n Approximate Hull with O( ) messages plus the costs of constructing a spanning tree. √nc c c • An k -Approximate Closest Pair on an [n ]×[n ] grid , for a constant c > 1/2, 2 plus the cost of computing a spanning tree, for any k ≤ n − 1. We also define a new version of the two-party communication problem, Path Com- putation, where two parties communicate through a path. We prove a lower bound on the communication complexity of this problem. v Table of Contents Supervisory Committee ii Abstract iii Table of Contents v List of Tables vii List of Figures viii Acknowledgements x 1 Introduction 1 1.1 Organization . 2 1.2 Preliminaries . 3 2 Red-Blue Geometric Covering 10 2.1 Introduction . 10 2.1.1 Preliminaries and Background . 12 2.1.2 Motivation and Discussion . 18 2.1.3 Sketch of the Techniques . 19 2.2 MaxRM-3SAT . 21 2.2.1 Hardness of Approximation MaxRM-3SAT . 21 2.3 Red-Blue Geometric Set Cover . 28 2.3.1 RBGSC[AARectangle] . 29 2.3.2 RBGSC[Convex] and RBGSC[Triangle] . 40 2.4 Boxes Class Cover . 43 2.4.1 Reduction from MaxRM-3SAT to BCC . 43 2.5 Conclusion and Open Problems . 46 vi 3 Distributed Geometric Networks 48 3.1 Preliminaries and Background . 49 3.1.1 Distributed Networks . 49 3.1.2 Geometric Networks . 54 3.1.3 Two-Party Communication . 58 3.1.4 Coreset . 60 3.2 Path Computation . 63 3.2.1 Path Computation Lower Bounds . 66 3.3 Diameter and Convex Hull . 70 3.3.1 -Kernel . 71 3.3.2 Approximate Diameter . 73 3.3.3 Approximate Convex Hull . 74 3.3.4 Lower Bound for Diameter . 76 3.3.5 Lower Bound for Convex Hull . 80 3.4 Closest Pair . 83 3.4.1 Approximate Closest Pair . 84 3.4.2 Lower Bound . 87 3.5 Conclusion and Open Problems . 91 Bibliography 94 vii List of Tables Table 2.1 Lower bounds shown in Chapter 2. * for any constant c < 1/2. 20 Table 3.1 Results of Chapter 3. Upper bound and lower bound of the num- ber of messages of size O(log n) bits for a geometric communica- tion network with n nodes on an [nc] × [nc] grid in the fixed-link model. *for a constant c > 1/2. ** for a constant c > 1 + 1/(2 lg n). 92 viii List of Figures Figure 2.1 Geometric Set Cover. (a) input for an instance of Geometric Set Cover (b) the set of the green objects is a solution for this instance. 13 Figure 2.2 Boxes Class Cover. (a) input for an instance of Boxes Class Cover, and (b) the set of green rectangles is a solution for this instance. 14 Figure 2.3 Red-Blue Geometric Set Cover. (a) input for an instance of Red-Blue Geometric Set Cover (b) the set of the green objects is a solution for this instance. 16 Figure 2.4 Reductions. −→∗ is a modified version of the reduction that Chan and Grant showed from SPECIAL-3SC to GSC[AARectangle] in [21]. −→∗∗ is the reduction that Bereg et al. showed from Rec- tilinear Polygon Covering to BCC in [11]. All of the reductions are the results of this dissertation. 19 Figure 2.5 It is not always possible to add a unique red point to each object. 29 Figure 2.6 Reduction from MaxRM-3SAT to RBGSC[AARectangle]. a) variable points for all the variables and four variable rectangles for X2 (b) Highlighted green areas are divisions of the plane to Region 1-3. c, d, e, f) clause points and clause rectangles ¯ for different types of clauses: c) (Xj ∨ Xl), d) (Xj ∨ Xl), e) ¯ ¯ ¯ ¯ (Xj ∨ Xl), f) (Xj ∨ Xj+1 ∨ Xj+2) . 32 Figure 2.7 Points and rectangles for Reduction from SPECIAL-3SC to RBGSC[AARectangle] . 40 Figure 2.8 points and convex shapes for the reduction from Set Cover to RBGSC[Convex] . 42 ix Figure 2.9 Reduction from MaxRM-3SAT to BCC. a) The points in the highlighted gray area are variable points for X2. b)Region 1-3. ¯ c, d, e, f) Added red points and blue points for c = (Xj ∨Xl), d) ¯ ¯ Added red points and blue points for (Xj ∨ Xl), c = (Xj ∨ Xl), ¯ ¯ c = (Xj ∨ Xj+1 ∨ Xj+2). ..................... 45 Figure 3.1 - kernel proof . 73 Figure 3.2 Reduction from Set Disjointness to Diameter. In this example, a1 = 1, ai = 0, ai+1 = 1, an = 1, b1 = 0, bi = 0, bi+1 = 1, and bn = 1. .............................. 77 Figure 3.3 The distance between wi and ui+1 is the second largest distance of a pair of points after the diameter. 80 Figure 3.4 Reduction from Set Disjointness to Convex Hull. In this exam- ple, a1 = 1, ai = 0, ai+1 = 1, an = 0, b1 = 1, bi = 0, bi+1 = 0, and bn = 1. .............................. 82 Figure 3.5 Reduction from Set Disjointness to Closest Pair. 88 Figure 3.6 Reduction from Set Disjointness to Closest Pair. In this ex- ample, a1 = 1, a2 = 1, an = 0, b1 = 0, b2 = 1, and bn = 0. ................................... 90 x ACKNOWLEDGEMENTS Many helped me on this journey, and I want to take a moment to thank them. First and foremost I would like to sincerely thank my supervisor, Dr. Valerie King, who has supported and advised me during my past few years with her knowledge and foresight. Valerie is an inspiring individual and supervisor. She has always motivated me to do my best in my research. I am grateful for all the technical skills and insights that she has shared and the engaging courses I took with her. I would like to thank both Dr. Venkatesh Sirvanasan and Dr. Will Evans, my supervisory committee members, for all the fruitful discussions and useful feedback I have received at every stage of my research. Venkatesh and Will were always very encouraging and helped me during hard times. I would like to thank Dr. Jit Bose, who agreed to be the external examiner and for all the insightful comments and suggestions.
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