SSStttooonnnyyy BBBrrrooooookkk UUUnnniiivvveeerrrsssiiitttyyy The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook University. ©©© AAAllllll RRRiiiggghhhtttsss RRReeessseeerrrvvveeeddd bbbyyy AAAuuuttthhhooorrr... Combinatorics and Complexity in Geometric Visibility Problems A Dissertation Presented by Justin G. Iwerks to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Applied Mathematics and Statistics (Operations Research) Stony Brook University August 2012 Stony Brook University The Graduate School Justin G. Iwerks We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Joseph S. B. Mitchell - Dissertation Advisor Professor, Department of Applied Mathematics and Statistics Esther M. Arkin - Chairperson of Defense Professor, Department of Applied Mathematics and Statistics Steven Skiena Distinguished Teaching Professor, Department of Computer Science Jie Gao - Outside Member Associate Professor, Department of Computer Science Charles Taber Interim Dean of the Graduate School ii Abstract of the Dissertation Combinatorics and Complexity in Geometric Visibility Problems by Justin G. Iwerks Doctor of Philosophy in Applied Mathematics and Statistics (Operations Research) Stony Brook University 2012 Geometric visibility is fundamental to computational geometry and its ap- plications in areas such as robotics, sensor networks, CAD, and motion plan- ning. We explore combinatorial and computational complexity problems aris- ing in a collection of settings that depend on various notions of visibility. We first consider a generalized version of the classical art gallery problem in which the input specifies the number of reflex vertices r and convex vertices c of the simple polygon (n = r + c). This additional information better char- acterizes the shape of the polygon. Through a lower bound construction, tight combinatorial bounds for coverage are achieved for all r 0 and c 3. ≥ ≥ The combinatorics of guarding polyominoes and other polyforms are stud- ied in terms of m, the number of cells, as opposed to the traditional parameter n. Various visibility models and guard types are considered. We establish that finding a minimum cardinality guard set for covering a polyomino is NP-hard. We introduce an algorithm for constructing a spiral serpentine polygoniza- tion of a set of n 3 points in the plane. The algorithm’s behavior can be viewed as incrementally≥ appending a visible triangle to the triangulation constructed so far. We consider beacon-based point-to-point routing and coverage problems. A beacon b is a point that can be activated to effect a gravitational pull toward itself in a polygonal domain. Algorithms are given for computing the attraction region of b and finding a minimum size set of beacons to route from a source s to a destination t given a finite set of candidate beacon locations. iii We show that finding a minimum cardinality set of beacons to cover a simple polygon or conduct certain types of routing in a simple polygon is NP-hard. iv Contents Abstract iii List of Figures vii Acknowledgments ix List of Publications x 1 Introduction 1 2 The Art Gallery Theorem for Simple Polygons in Terms of theNumberofReflexandConvexVertices 5 2.1 Introduction............................ 5 2.2 PreviousWork .......................... 6 2.3 LowerBoundConstructions . 7 2.4 Conclusion............................. 13 3 Guarding Polyominoes 14 3.1 Introduction............................ 14 3.2 CombinatorialBounds . 17 3.2.1 NecessaryConditions . 17 3.2.2 SufficiencyConditions . 18 3.2.3 GeneralizationtoRectanglominoes . 25 3.3 Hardness.............................. 26 3.3.1 From MAX2SAT(2L) to MLCP in Octagonal Grids . 27 3.3.2 FromMLCPtoPolyominoes. 30 3.4 AlgorithmsforSpecialCases. 33 3.4.1 GuardingThinPolyominoTrees . 34 3.4.2 GuardingThinPolyominoPaths . 34 3.4.3 ProofsofOptimality . 36 v 3.5 Conclusion.............................. 41 4 Guarding Polyforms 42 4.1 Introduction............................ 42 4.2 NecessaryConditions . 43 4.2.1 Polycubes ......................... 43 4.2.2 Polyiamonds........................ 44 4.2.3 Polyhexes ......................... 45 4.3 SufficiencyConditions . 46 4.3.1 Polycubes ......................... 46 4.3.2 Polyiamonds........................ 59 4.3.3 Polyhexes ......................... 68 4.4 Conclusion............................. 72 5 Spiral Serpentine Polygonization of a Planar Point Set 73 5.1 Introduction............................ 73 5.2 TheAlgorithm .......................... 75 5.3 Correctness ............................ 78 5.4 ImplementationandExamples . 81 5.5 Conclusion.............................. 81 6 Beacon-BasedRoutingandCoverage 83 6.1 Introduction............................ 83 6.1.1 OurResults ........................ 84 6.2 Beacon-BasedRouting . 85 6.2.1 Combinatorics of All Pairs Routing in a Simple Polygon 85 6.2.2 Computing the Attraction Region of a Beacon . 90 6.2.3 Routing with a Discrete Set of Candidate Beacons . 97 6.3 HardnessofBeaconCoverage . 98 6.4 Conclusion............................. 99 References 100 Appendix A 105 vi List of Figures 2.1 ShutterPolygons ......................... 6 2.2 Guarding Pseudotriangle Chain Polygons . 7 2.3 DoubleSweepTechnique . 8 2.4 PseudotriangleChainLabeling. 9 2.5 Illustration of Lower Bound Constructions . 12 3.1 AnExampleofaPolyomino . 15 3.2 VisibilityModels ......................... 16 3.3 PointGuardLowerBoundConstruction . 17 3.4 PixelGuardLowerBoundConstruction . 18 3.5 CaseAnalysisforCoveringSmallPolyominoes . 19 3.6 Special5-Polyominoes . 20 3.7 BFSTreeStructure......................... 21 3.8 Finding a Subtree That Forms a Good Polyomino . 22 3.9 OpenPixelGuardLowerBoundConstruction . 24 3.10 Differing Guard Numbers Between a Rectanglomino and its As- sociatedPolyomino ........................ 25 3.11 VariableGadget.......................... 28 3.12 NestingBoxesforPlacingClauseGadgets . 28 3.13ClauseGadgets .......................... 29 3.14 HorizontalRayGadget . 30 3.15 Slant-RayGadget. 31 3.16RayGadgetsonBox ....................... 32 3.17 FourFreeSubpolyominoTypes . 37 3.18 NineFreeSubpolyominoTypes . 40 4.1 LowerBoundConstructionsforPolycubes . 44 4.2 LowerBoundConstructionsforPolyiamonds . 45 4.3 LowerBoundConstructionsforPolyhexes . 45 4.4 Special5-polycubes . 47 vii 4.5 An8-PolycubeRequiring3PointGuards . 49 4.6 An11-PolycubeRequiring4PointGuards . 50 4.7 ExampleCasesthatAreNotVoxel-Good . 51 4.8 MoreExamplesCasesthatAreNotVoxel-Good . 52 4.9 A23-PolycubeRequiring6PointGuards . 54 4.10 Special6-Polycubes. 55 4.11 AnExampleIllustratingSubcase3h. 59 4.12 The 15 free 8-Polyiamonds Having a Dual that is a Path . 61 4.13 Analysis for Case (5e) of Guarding Polyiamonds with Point Guards............................... 64 4.14 10-Polyiamonds with Path Duals Requiring 2 Guards . 65 4.15 Illustrations of Pixel Guard Placement for Case (6b) . .... 67 4.16 TheSevenFree4-Polyhexes . 69 4.17 Team-upCoverageofaPixelinaPolyhex . 71 5.1 Spiral Polygon and a Serpentine Triangulation . 74 5.2 SpiralingPolygonalPath . 75 5.3 Two Cases Arising During Algorithm Execution . 78 ′ 5.4 v vqk+1 CannotFormaRightTurn . 79 5.5 ExperimentalResults. 82 6.1 DeadPoints............................ 84 6.2 Lower Bound Construction for Beacon Point-to-Point Routing 85 6.3 BeaconPlacementCaseAnalysis . 86 6.4 InductivePlacementofBeacons . 87 6.5 ZigzagSpikeGadget ....................... 89 6.6 AllDestinationsSpikeGadget . 90 6.7 Upper Bound on the Number of Dead Points in a Simple Polygon 93 6.8 Cut-VertexClasses ........................ 94 6.9 ArrowSpikeGadget ....................... 98 A.1 TheCasesforaSubtreeofHeight3 . 108 A.2 TheSpecialCoverforSubtreesofSize7 . 110 A.3 TheCasesforaSubtreeofHeight4 . 114 A.4 TheCasesforaSubtreeofHeight5 . 116 viii Acknowledgments I am indebted to my advisor, Joseph S. B. Mitchell, for helping me develop into an independent mathematical researcher. I must also thank Joe for introducing me to the wonderful world of computational geometry. I hope to inspire my own students with the rich intensity of excitement and passion he has for this field. I would like to thank James Glimm and Xiaolin Li for their valuable guidance and support while I began my first research projects. I am grateful for my colleagues and co-authors including Esther Arkin, Therese Biedl, Michael Biro, Brian Fix, Jie Gao, Mohammed T. Irfan, Tulin Ka- man, Ryan Kaufman, Joondong Kim, Irina Kostitsyna, Hyunkyung Lim, Alan Tucker, Shang Yang and Yan Yu. I would like to thank Amy Picard Winston, my superb high school physics teacher, for inspiring me to pursue physics and mathematics in college. I am thankful for my parents and siblings who always made learning fun growing up and encouraged me to pursue my academic interests. I would like to thank my wife, Rebecca, for her steadfast patience and support. Despite moving three times, years of long work commutes, and budgeting with a graduate student’s income, she has gracefully stuck by my side. Thanks also to our daughter, Cambria, for providing uncountable amounts of joy. Completing my doctoral degree is nothing short of a grand and improbable victory that I could never have accomplished alone. I therefore wish to give thanks to Jesus Christ and close with the following quote: Then Samuel took a stone and set it up between Mizpah and Shen. He named it Ebenezer, saying, “Thus