Euclidean Shortest Paths “Beauty on the Path”, a digital painting by Stephen Li (Auckland, New Zealand), September 2011, provided as a gift for this book. Fajie Li r Reinhard Klette Euclidean Shortest Paths Exact or Approximate Algorithms Fajie Li Reinhard Klette School of Information Science Dept. Computer Science and Technology University of Auckland Huaqiao University P.O. Box 92019 P.O. Box 800 Auckland 1142 Xiamen Fujian New Zealand People’s Republic of China [email protected] [email protected] ISBN 978-1-4471-2255-5 e-ISBN 978-1-4471-2256-2 DOI 10.1007/978-1-4471-2256-2 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: 2011941219 © Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as per- mitted 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 publish- ers, 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 (www.springer.com) To Zhixing Li, and to the two youngest in the Klette family in New Zealand 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 short- est 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 obsta- cles 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 dif- fers 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 prob- lems. 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 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 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 or- der, 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. Sei- del’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 al- gorithm 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 cal- culating 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 al- gorithms 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 x Preface proofs, the analysis of time complexities and related topics, and not just the algo- rithms 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 exer- cises, 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 de- scribed projects do not exist, leaving space for the creativity of the student. The audience for the book could be students in computer science, IT, mathemat- ics, or engineering at a university, or academics being involved in research or teach- ing of efficient algorithms. The book could also be useful for programmers, mathe- maticians, 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, Chander- jit 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 Rosen- feld, the late Klaus Voss, Jinlong Wang, and Joviša Žunic´ for discussions or com- ments that were of relevance for this book. The authors thank Chengle Huang (ChingLok Wong) for discussions on rubber- band 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 Algo- rithm 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.