Euclidean Shortest Paths

Transcription

1 Euclidean Shortest Paths

2 Beauty on the Path, a digital painting by Stephen Li (Auckland, New Zealand), September 2011, provided as a gift for this book.

3 Fajie Li Reinhard Klette Euclidean Shortest Paths Exact or Approximate Algorithms

4 Fajie Li School of Information Science and Technology Huaqiao University P.O. Box 800 Xiamen Fujian People s Republic of China li.fajie@yahoo.com Reinhard Klette Dept. Computer Science University of Auckland P.O. Box Auckland 1142 New Zealand r.klette@auckland.ac.nz ISBN e-isbn DOI / Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To Zhixing Li, and to the two youngest in the Klette family in New Zealand

6 Foreword The world is continuous the mind is discrete. David Mumford (born 1937) Recently, I was confronted with the problem of planning my travel from Israel to New Zealand, home of the two authors of this book. When taking two antipodal points on the globe, like Haifa and Queenstown, there is an infinite number of shortest paths connecting these points. Still, due to constraints like reachable airports and airlines, finding the optimal solution was almost immediate. Throughout the long history of geometry sciences, the problem of finding the shortest path in various scenarios occupied the minds of researchers in many fields. Even in Euclidean spaces, which are considered simple, the introduction of obstacles leads to challenging problems for which efficient computational solvers are hard to find. The optimal path in 3D space with polyhedral obstacles was among the first geometric problems proven to be, at least formally, computationally hard to solve. It took almost 20 years for a team of 5 programming experts to eventually implement a method approximating the continuous Dijkstra algorithm that is reviewed in this book. Exact problems are hard to solve, and approximations are obviously required. My personal line of work when dealing with geometric problems somewhat differs from the school of thought promoted by this book. A numerical approximation in my vocabulary involves the notion of accuracy that depends on an underlying grid resolution. This grid is defined by sampling the domain of the problem and leads to the field of numerical geometry in which efficient solvers are simple to design. The alternative computational geometry school of thought describes obstacles as polyhedral structures that allegedly define the exact problem. The resulting challenges under this setting are extremely difficult to overcome. Still, the unifying bridge between these two philosophical branches is defined by the geometric problems. Without being familiar with the difficulty involved in designing a path between points in a weighted domain, one could not appreciate the conceptual simplicity of numerical Eikonal solvers. This book addresses the type of hard problems in the computational geometry flavor while inventing constraints that allow for efficient solvers to be designed. For example, the creative rubberband methods explored in this book restrict the optimal vii

7 viii Foreword paths to bands of bounded width, thereby redefining problems and simplifying the challenges, proving yet again Aleksandr Pushkin s observation that inspiration is needed in geometry, just as much as in poetry. I hope that, like me, the reader would find the geometrical challenges introduced in this book fascinating and also appreciate the elegance of the proposed solutions. Haifa, Israel Ron Kimmel

8 Preface A Euclidean shortest path connects a source with a destination, avoids some places (called obstacles), visits some places (called attractions), possibly in a defined order, and is of minimum length. Euclidean shortest-path problems are defined in the Euclidean plane or in Euclidean 3-dimensional space. The calculation of a convex hull in the plane is an example for finding a shortest path (around the given set of planar obstacles). Polyhedral obstacles and polyhedral attractions, a start and an endpoint define a general Euclidean shortest-path problem in 3-dimensional space. The book presents selected algorithms (i.e., not aiming at a general overview) for the exact or approximate solution of shortest-path problems. Subjects in the first chapters of the book also include fundamental algorithms. Graph theory offers shortest-path algorithms for discrete problems. Convex hulls (and to a lesser extent also constrained convex hulls) have been discussed in computational geometry. Seidel s triangulation and Chazelle s triangulation method for a simple polygon, and Mitchell s solution of the continuous Dijkstra problem have also been selected for a detailed presentation, just to name three examples of important work in the area. The book also covers a class of algorithms (called rubberband algorithms), which originated from a proposal for calculating minimum-length polygonal curves in cube-curves; Thomas Bülow was a co-author of the initiating publication, and he coined the name rubberband algorithm in 2000 for the first time for this approach. Subsequent work between 2000 and now shows that the basic ideas of this algorithm generalised for solving a range of problems. In a sequence of publications between 2003 and 2010, we, the authors of this book, describe a class of rubberband algorithms with proofs of their correctness and time-efficiency. Those algorithms can be used to solve different Euclidean shortest-path (ESP) problems, such as calculating the ESP inside of a simple cube-arc (the initial problem), inside of a simple polygon, on the surface of a convex polytope, or inside of a simple polyhedron, but also ESP problems such as touring a finite sequence of polygons, cutting parts, or the safari, zookeeper, or watchman route problems. We aimed at writing a book that might be useful for a second or third-year algorithms course at the university level. It should also contain sufficient details for students and researchers in the field who are keen to understand the correctness ix

9 x Preface proofs, the analysis of time complexities and related topics, and not just the algorithms and their pseudocodes. The book discusses selected subjects and algorithms at some depth, including mathematical proofs for most of the given statements. (This is different from books which aim at a representative coverage of areas in algorithm design.) Each chapter closes with theoretical or programming exercises, giving students various opportunities to learn the subject by solving problems or doing their own experiments. Tasks are (intentionally) only sketched in the given programming exercises, not described exactly in all their details (say, as it is typically when a costumer specifies a problem to an IT consultant), and identical solutions to such vaguely described projects do not exist, leaving space for the creativity of the student. The audience for the book could be students in computer science, IT, mathematics, or engineering at a university, or academics being involved in research or teaching of efficient algorithms. The book could also be useful for programmers, mathematicians, or engineers which have to deal with shortest-path problems in practical applications, such as in robotics (e.g., when programming an industrial robot), in routing (i.e., when selecting a path in a network), in gene technology (e.g., when studying structures of genes), or in game programming (e.g., when optimising paths for moves of players) just to cite four of such application areas. The authors thank (in alphabetical order) Tetsuo Asano, Donald Bailey, Chanderjit Bajaj, Partha Bhowmick, Alfred (Freddy) Bruckstein, Thomas Bülow, Xia Chen, Yewang Chen, David Coeurjolly, Eduardo Destefanis, Michael J. Dinneen, David Eppstein, Claudia Esteves Jaramillo, David Gauld, Jean-Bernard Hayet, David Kirkpatrick, Wladimir Kovalevski, Norbert Krüger, Jacques-Olivier Lachaud, Joe Mitchell, Akira Nakamura, Xiuxia Pan, Henrik G. Petersen, Nicolai Petkov, Fridrich Sloboda, Gerald Sommer, Mutsuhiro Terauchi, Ivan Reilly, the late Azriel Rosenfeld, the late Klaus Voss, Jinlong Wang, and Joviša Žunić for discussions or comments that were of relevance for this book. The authors thank Chengle Huang (ChingLok Wong) for discussions on rubberband algorithms; he also wrote C++ programs for testing Algorithms 7 and 8. We thank Jinling Zhang and Xinbo Fu for improving C++ programs for testing Algorithm 7. The authors acknowledge computer support by Wei Chen, Wenze Chen, Yongqian Du, Wenxian Jiang, Yanmin Luo, Shujuan Peng, Huijuan Pi, Huazhen Wang, and Jian Yu. The first author thanks dean Weibin Chen at Huaqiao University for supporting the project of writing this book. The second author thanks José L. Marroquín at CIMAT Guanajuato for an invitation to this institute, thus providing excellent conditions for working on this book project. Parts of Chap. 4 (on relative convex hulls) are co-authored by Gisela Klette, who also contributed comments, ideas and criticisms throughout the book project. We are grateful to Garry Tee for corrections and valuable comments, often adding important mathematical or historic details. Huaqiao, People s Republic of China Auckland, New Zealand Fajie Li Reinhard Klette

10 Contents Part I Discrete or Continuous Shortest Paths 1 Euclidean Shortest Paths ArithmeticAlgorithms Upper Time Bounds FreeParametersinAlgorithms AnUnsolvableProblem Distance, Metric, and Length of a Path A Walk in Ancient Alexandria Shortest Paths in Weighted Graphs Points, Polygons, and Polyhedra in Euclidean Spaces Euclidean Shortest Paths Problems Notes References Deltas and Epsilons Exact and δ-approximatealgorithms ApproximateIterativeESPAlgorithms Convergence Criteria Convex Functions Topology in Euclidean Spaces Continuous and Differentiable Functions; Length of a Curve Calculating a Zero of a Continuous Function Cauchy s Mean-Value Theorem Problems Notes References Rubberband Algorithms PursuitPaths Fixed or Floating ESP Problems; Sequence of Line Segments xi

11 xii Contents 3.3 Rubberband Algorithms A Rubberband Algorithm for Line Segments in 3D Space Asymptotic and Experimental Time Complexity ProofofCorrectness Processing Non-disjoint Line Segments as Inputs More Experimental Studies An Interesting Input Example of Segments in 3D Space A Generic Rubberband Algorithm Problems Notes References Part II Paths in the Plane 4 Convex Hulls in the Plane ConvexSets Convex Hull and Shortest Path; Area ConvexHullofaSetofPointsinthePlane Convex Hull of a Simple Polygon or Polyline RelativeConvexHulls Minimum-Length Polygons in Digital Pictures Relative Convex Hulls The General Case Problems Notes References Partitioning a Polygon or the Plane Partitioning and Shape Complexity Partitioning of Simple Polygons and Dual Graphs Seidel s Algorithm for Polygon Trapezoidation Inner, Up-, Down-, or Monotone Polygons Trapezoidation of a Polygon at Up- or Down-Stable Vertices Time Complexity of Algorithm Polygon Trapezoidation Method by Chazelle The Continuous Dijkstra Problem Wavelets and Shortest-Path Maps Mitchell salgorithm Problems Notes References ESPs in Simple Polygons Properties of ESPs in Simple Polygons Decompositions and Approximate ESPs Chazelle Algorithm TwoApproximateAlgorithms Chazelle Algorithm Versus Both RBAs...180

12 Contents xiii Part III 6.6 Turning the Approximate RBA into an Exact Algorithm Problems Notes References Paths in 3-Dimensional Space 7 Paths on Surfaces Obstacle Avoidance Paths in 3D Space Polygonal Cuts and Bands ESPs on Surfaces of Convex Polyhedrons ESPs on Surfaces of Polyhedrons The Non-existence of Exact Algorithms for Surface ESPs Problems Notes References Paths in Simple Polyhedrons Types of Polyhedrons; Strips ESPComputation TimeComplexity Examples:ThreeNP-CompleteorNP-HardProblems Conclusions for the General 3D ESP Problem Problems Notes References Paths in Cube-Curves The Cuboidal World Original and Revised RBA for Cube-Curves AnAlgorithmwithGuaranteedErrorLimit MLPs of Decomposable Simple Cube-Curves AnalysisoftheOriginalRBA RBAs for MLP Calculation in Any Simple Cube-Curve CorrectnessProof Time Complexities and Examples The Non-existence of Exact Solutions Problems Notes References Part IV Art Galleries 10 Touring Polygons About TPP Contributions in This Chapter TheAlgorithms...317

13 xiv Contents 10.4 Experimental Results Concluding Remarks and Future Work Problems Notes References Watchman Routes EssentialCuts Algorithms CorrectnessandTimeComplexity Problems Notes References Safari and Zookeeper Problems FixedandFloatingProblems;Dilations SolvingtheSafariRouteProblem Solving the Zookeeper Route Problem Some Generalisations Problems Notes References Appendix Mathematical Details A.1 Derivatives for Example A.2 GAP Inputs and Outputs A.3 Matrices Q for Sect Index...371

14 Abbreviations Definitions 2D 2-dimensional 3D 3-dimensional AMLPP Approximate minimum-length pseudo-polygon; see page 276 CAV Cavity; see page 98 CH Convex hull; see page 97 ESP Euclidean shortest path; see page 11 iff Read: if and only if; see page 6 MLA Minimum-length arc; see page 279 MLP Minimum-length polygon; see page 114, , 304 MLPP Minimum-length pseudo-polygon; see page 275 MPP Minimum-perimeter polygon; see page 111 RBA Rubberband algorithm; see page 58 RCH Relative convex hull; see page 121 SPM Shortest path map; see page 159 SRP Safari route problem; see page 349 TPP Touring polygon problem; see page 313 WRP Watchman route problem; see page 327 ZRP Zookeeper route problem; see page 349 Symbols S S pq pqr Cardinality of set S Relation sign for being parallel Frontier of set S Logical and Logical or Intersection of sets Union of sets Straight line segment with endpoints p and q Triangle with vertices p, q, and r Trapezoid or triangle (in a partitioning) xv

15 xvi Abbreviations pq Straight line defined by points p and q; orientation from p to q (pqr) Angle formed by rotating segment pq clockwise into segment qr End of a proof or of an example a,b,c Real numbers A( ) Area of a measurable set (as a function) α, β Angles C Set of complex numbers a + i b, with i = 1 and a,b R d m Minkowski metrics, for m = 1, 2,...or m = d e Euclidean metric; note that d e = d 2 D Determinant; see page 96 δ Real number greater than zero e Edge (e.g., of a graph); real constant e = exp(1) E Set of edges of a graph ε Real number greater than zero (the accuracy of an RBA) ε s Real number greater than zero (a shift distance) f,g,h Functions, for example from N into R g Simple cube-curve g Tube (i.e., the union of cubes) of a simple cube-curve g F Face of a polyhedron F Set of faces of a polyhedron G Graph G Set {..., 2, 1, 0, 1, 2,...} of integers γ Curve in Euclidean space (e.g., straight line, polyline, smooth curve) H Half plane; see page 95 i, j, k, l, m, n Natural numbers i Number of iterations, e.g., in a rubberband algorithm j Natural number; index of points or vertices in a path k Natural number; total number of items κ Function in ε L Length (as a real number) L( ) Length of a rectifiable curve (as a function) l(ρ) Length of a path ρ λ Real number; e.g., between 0 and 1 n Natural number; e.g., defining the complexity of the input N Neighbourhood (in the Euclidean topology, or in grids) N Set {0, 1, 2,...} of natural numbers O( ) Asymptotic upper time bound p,q Points in R 2 or R 3, with coordinates x p, y p,orz p p(x) Polynomial in x p.x,p i.x x-coordinate of point p or point p i p.y,p i.y y-coordinate of point p or point p i p.z,p i.z z-coordinate of point p or point p i P Polygon

16 Abbreviations xvii P, P Closure and topological interior of polygon P π Plane in R 3 ; real constant π = 4 arctan(1) Polyhedron, Closure and topological interior of polyhedron ki=1 S i Product of sets S i P Partitioning (of the plane or of a simple polygon) r Radius of a disk or sphere; point in R 2 or R 3 R Set of real numbers ρ Path with finite number of vertices; see page 11 s Point in R 2 or R 3 S,S i Sets S Family of sets t Point in R 2 or R 3 T Tree; threshold (a real number) τ Threshold (real number) T Trapezoidation or triangulation (of the plane or of a simple polygon) v Vertex or node; a point in R 2 or R 3, with coordinates x v, y v,orz v V Set of vertices of a graph V(G),V(T) Set of vertices of graph G or tree T V(ρ) Set of vertices of a path ρ vu, vu Undirected or directed line segment between points v and u x Real variable x Vector y Real variable y Vector Z Set of integers