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Stony Brook University 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... Some Path Planning Algorithms in Computational Geometry and Air Traffic Management A Dissertation Presented by Shang Yang to The Graduate School in Partial fulfillment of the Requirements for the Degree of Doctor of Philosophy in Computer Science Stony Brook University August 2012 Stony Brook University The Graduate School Shang Yang We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Dissertation Adviser { Professor Joseph S. B. Mitchell Department of Applied Mathematics and Statistics Chairperson of Defense { Associate Professor Jie Gao Computer Science Department Professor Esther M. Arkin Department of Applied Mathematics and Statistics Assistant Professor Jiaqiao Hu Department of Applied Mathematics and Statistics This dissertation is accepted by the Graduate School. Charles Taber Interim Dean of the Graduate School ii Abstract of the Dissertation Some Path Planning Algorithms in Computational Geometry and Air Traffic Management by Shang Yang Doctor of Philosophy in Computer Science Stony Brook University 2012 Computing optimal routes subject to geometric constraints is a fundamental area of research in computational geometry. This thesis is devoted to the study of some specific geometric routing and optimization problems. The first main area of the thesis addresses a class of multicommodity flow problems in geometric domains: For a given planar domain P populated with obstacles of different types, we consider routing thick paths corresponding to motion of vehicles of different classes, from a source edge on the boundary of P to a sink edge on the boundary of P. Each class of vehicle has an associated path width required and a given set of types of obstacles it must avoid, while it can freely pass through other types of obstacles. The problem arises in air traffic management with different classes of aircraft that must avoid various types of weather disturbances. We show that the decision problem is NP-hard even when there are only two classes of vehicles and two types of obstacles. We present approximation algorithms for the multicriteria optimization problems that arise when trying to maximize the number of routable paths. We also give heuristics and provide an experimental analysis of their effectiveness. The second problem we address is that of computing a path or cycle that is constrained to be convex and to intersect a given set of connected, compact sets. In particular, we resolve an open problem posed more than two decades ago by Arik Tamir: Given a collection of compact sets, can one efficiently determine if there is a convex body whose boundary intersects every set in the collection? We prove that it is NP-hard, in general, to decide the existence of a convex traversal path, even if the input is a set of line segments in the plane. Further, we generalize our proof to show that deciding the existence of a convex surface stabbing a set of balls in three dimensions is NP-hard. On the positive side, we give a polynomial-time algorithm to find a convex transversal of a iii maximum number of pairwise-disjoint segments in 2D, assuming the vertices of the transversal are restricted to a given set of points. In the third part of the thesis, we investigate the problem of computing paths and trees within an uncertain geometric domain, in which the set of obstacles is not known deterministically, but is specified by a stochastic model. The problems arise in routing aircraft through uncertain weather systems, especially in the case of merging flows of aircraft arriving to a terminal airspace in the presence of uncertain events that impact the availability of airspace. We formalize the problem of computing a highly probably path in geometric settings and prove NP-hardness of the general problem. Further, we develop efficient algorithms for computing paths and trees in the presence of uncertain geometric obstacles. We apply our methods to the problem of computing routes and trees for flows of aircraft that are designed to be robust to off-nominal events that may impact the super-dense operations airspace through which they are routed. iv Contents List of Figures . ix Acknowledgements . .x 1 Introduction 1 2 Routing Multi-Class Traffic Flows in the Plane 3 2.1 Introduction . .4 2.1.1 Motivation . .4 2.1.2 Related Work . .6 2.1.3 Summary of Contributions . .6 2.2 Problem Formulation and Overview of the Results . .7 2.2.1 Type Sequence and Uppermost Paths . .7 2.3 Testing type sequence feasibility in the monochromatic case . 11 2.4 Hardness results . 13 2.4.1 Only blue obstacles . 17 2.4.2 Hardness of approximation . 17 2.4.3 Hardness of the Two Widths paths problem . 18 2.5 Approximation . 19 2.6 Small number of holes . 22 2.7 Practical Heuristics: Implementation and Experiments . 24 2.7.1 Data Sets . 26 2.7.2 Enumeration of Type Sequences . 26 2.7.3 A Heuristic for Short Paths . 28 2.7.4 Experimental Results . 29 2.8 Conclusion . 30 3 Flexible Air Lane Generation to Maximize Flow Under Hard and Soft Constraints 32 3.1 Introduction . 33 v 3.1.1 Related Work . 35 3.1.2 Chapter Organization . 36 3.2 Modeling . 36 3.2.1 Airspace Model . 37 3.2.2 Hard and Soft Constraints . 38 3.2.3 Objective . 39 3.3 Maximum Flow Rate Theory . 40 3.3.1 Flows in Discrete Networks . 40 3.3.2 Multi-Class Throughput Problem Formulation . 42 3.3.3 Class Sequences . 43 3.3.4 Bottommost Paths . 43 3.3.5 Theoretical Results . 43 3.4 Algorithmic Solution Approaches . 44 3.4.1 Bottommost Path Filling Algorithm . 44 3.4.2 Postprocessing: Tautening Bottommost Paths . 47 3.5 Experiments . 47 3.5.1 Real Weather Data . 47 3.5.2 Experimental Results . 48 3.5.3 Probabilistic Weather Maps . 49 3.6 Conclusion . 51 3.7 Future Research . 52 4 Convex Transversals 54 4.1 Introduction . 55 4.1.1 Contributions . 55 4.1.2 Closed stabbers vs. Terrains . 56 4.2 Hardness results . 56 4.2.1 Stabbing segments in the plane is NP-hard . 56 4.2.2 Stabbing squares and scaled copies of a convex polygon 59 4.2.3 Stabbing balls in 3D is NP-hard . 62 4.3 Stabbing disjoint segments . 70 4.4 Stabbing with vertices of a regular polygon . 77 4.4.1 The decision problem . 77 4.4.2 Optimization problem: Symmetry with imprecision . 78 4.5 Conclusion . 81 5 Robust Trees and Highly Probable Path Problem 83 5.1 Problem Formulation and Results . 84 5.1.1 Related Work . 85 vi 5.1.2 Hardness Results for the Segment Case . 86 5.1.3 Exploring the Disk Case . 89 5.2 Practical Heuristics: Implementation . 91 5.2.1 Grid Generation . 92 5.2.2 Search Graph Generation and Test Graph against Weather Constraints . 95 5.2.3 Phase 1 Tree Computation . 97 5.2.4 Phase 2 Tree Optimization . 99 5.2.5 Operational Flexibility . 100 5.2.6 Illustrations . 103 5.3 Practical Heuristics: Experiments . 103 5.3.1 Speed Tests . 104 5.3.2 Robustness Tests . 105 6 Open Problems and Future Research 109 Bibliography 111 vii List of Figures 2.1 An example of the multi-class routing problem . .5 2.2 An example of uppermost paths . .8 2.3 The critical graph of a polygonal domain . 13 2.4 Hardness gadgets illustration . 14 2.5 An example of the hardness reduction . 15 2.6 Correctness of vertex gadgets . 16 2.7 An example of the 2-width path routing problem hardness re- duction . 18 2.8 Approximation . 21 2.9 Pareto Frontier . 21 2.10 Transforming bundles into thick paths . 22 2.11 Creating a polygonal domain from a 3-PARTITION instance . 23 2.12 Reducing the number of color swaps . 24 2.13 Routing bottommost paths in the grid . 25 2.14 An example of a real weather test case . 27 2.15 Examples of routing and Pareto frontier . 27 2.16 Examples of routing in a randomly generated test case . 28 3.1 Mapping real weather obstacles into geometric constraints . 37 3.2 Airlane packing . 38 3.3 Min-Cut in geometric domains . 41 3.4 A thick path . 42 3.5 The underlying graph used in bottommost filling . 45 3.6 An example of bottommost filling . 46 3.7 Real weather experiments . 49 3.8 Probabilistic weather maps experiments . 51 3.9 Synthesized weather data experiments . 52 4.1 Transversals v.s. terrains . 57 4.2 Hardness proof for stabbing segments . 58 viii 4.3 Hardness proof for stabbing squares . 60 4.4 An example of the hardness gadgets construction . 61 4.5 Variable hardness gadgets for stabbing balls in 3D . 63 4.6 Clause gadgets for stabbing balls in 3D . 65 4.7 An overview of the hardness proof for stabbing balls in 3D . 66 4.8 Correctness of the hardness proof (1) . 69 4.9 Correctness of the hardness proof (2) . 71 4.10 Correctness of the hardness proof (3) . 72 4.11 Arcs, bridges and chords . 73 4.12 A segment-free triangle . 74 4.13 Finding the segment-free triangle . 76 4.14 The dynamic programming algorithm recursion . 77 4.15 Symmetry with imprecision . 80 5.1 Highly probable path problem . 85 5.2 Vertex gadgets construction . 86 5.3 Edge gadgets construction . 87 5.4 An example of the hardness construction .
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