Euclidean Shortest Paths - Semantic Scholar

Fajie Li and Reinhard Klette

Euclidean Shortest Paths Exact or Approximate Algorithms

Springer

Contents

Foreword by Ron Kimmel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Part I Discrete or Continuous Shortest Paths 1

Euclidean Shortest Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Arithmetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Upper Time Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Free Parameters in Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 An Unsolvable Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Distance, Metric, and Length of a Path . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 A Walk in Ancient Alexandria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Shortest Paths in Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Points, Polygons, and polyhedra in Euclidean Spaces . . . . . . . . . . . . . 1.9 Euclidean Shortest Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 14 15 16 20 23 27 31 34 36

2

Deltas and Epsilons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Exact and δ -Approximate Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Approximate Iterative ESP Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Topology in Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Continuous and Differentiable Functions; Length of a Curve . . . . . . 2.7 Calculating the Zero of a Continuous Function . . . . . . . . . . . . . . . . . . 2.8 Cauchy’s Mean-Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 42 44 47 48 50 53 55 56 58

3

4

3

Contents

Rubberband Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Pursuit Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fixed or Floating ESP Problems; Sequence of Line Segments . . . . . 3.3 Rubberband Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 A Rubberband Algorithm for Line Segments in 3D Space . . . . . . . . . 3.5 Asymptotic and Experimental Time Complexity . . . . . . . . . . . . . . . . . 3.6 Proof of Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Processing Non-Disjoint Line Segments as Inputs . . . . . . . . . . . . . . . 3.8 More Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 An Interesting Input Example of Segments in 3D Space . . . . . . . . . . 3.10 A Generic Rubberband Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 61 63 65 66 69 75 78 80 81 91 93

Part II Paths in the Plane 4

Convex Hulls in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Convex Hull and Shortest Path; Area . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3 Convex Hull of a Set of Points in the Plane . . . . . . . . . . . . . . . . . . . . . 104 4.4 Convex Hull of a Simple Polygon or Polyline . . . . . . . . . . . . . . . . . . . 109 4.5 Relative Convex Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.6 Minimum-Length Polygons in Digital Pictures . . . . . . . . . . . . . . . . . . 117 4.7 Relative Convex Hulls - The General Case . . . . . . . . . . . . . . . . . . . . . . 119 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5

Partitioning a Polygon or the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.1 Partitioning and Shape Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2 Partitioning of Simple Polygons and Dual Graphs . . . . . . . . . . . . . . . 131 5.3 Seidel’s Algorithm for Polygon Trapezoidation . . . . . . . . . . . . . . . . . . 134 5.4 Inner, Up-, Down-, or Monotone Polygons . . . . . . . . . . . . . . . . . . . . . 140 5.5 Trapezoidation of a Polygon at Up- or Down-Stable Vertices . . . . . . 143 5.6 Time Complexity of Algorithm 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.7 Polygon Trapezoidation Method by Chazelle . . . . . . . . . . . . . . . . . . . 155 5.8 The Continuous Dijkstra Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.9 Wavelets and Shortest-Path Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.10 Mitchell’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6

ESPs in Simple Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.1 Properties of ESPs in Simple Polygons . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2 Decompositions and Approximate ESPs . . . . . . . . . . . . . . . . . . . . . . . 175 6.3 Chazelle Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.4 Two Approximate Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Contents

5

6.5 Chazelle Algorithm versus both RBAs . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.6 Turning the Approximate RBA into an Exact Algorithm . . . . . . . . . . 183 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Part III Paths in 3-Dimensional Space 7

Paths on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.1 Obstacle Avoidance Paths in 3D Space . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.2 Polygonal Cuts and Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.3 ESPs on Surfaces of Convex Polyhedrons . . . . . . . . . . . . . . . . . . . . . . 195 7.4 ESPs on Surfaces of Polyhedrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.5 The Non-Existence of Exact Algorithms for Surface ESPs . . . . . . . . 205 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8

Paths in Simple Polyhedrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.1 Types of Polyhedrons; Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.2 ESP Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.3 Time Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.4 Examples: Three NP-complete or NP-hard Problems . . . . . . . . . . . . . 222 8.5 Conclusions for the General 3D ESP Problem . . . . . . . . . . . . . . . . . . . 224 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

9

Paths in Cube-Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.1 The Cuboidal World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9.2 Original and Revised RBA for Cube-Curves . . . . . . . . . . . . . . . . . . . . 234 9.3 An Algorithm with Guaranteed Error Limit . . . . . . . . . . . . . . . . . . . . . 237 9.4 MLPs of Decomposable Simple Cube-Curves . . . . . . . . . . . . . . . . . . . 241 9.5 Analysis of the Original RBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 9.6 RBAs for MLP Calculation in Any Simple Cube-Curve . . . . . . . . . . . 272 9.7 Correctness Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 9.8 Time Complexities and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 9.9 The Non-Existence of Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 287 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Part IV Art Galleries 10

Touring Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 10.1 About TPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 10.2 Contributions in this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 10.3 The Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 10.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 10.5 Concluding Remarks and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 318

6

Contents

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 11

Watchman Routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 11.1 Essential Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 11.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11.3 Correctness and Time Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

12

Safari and Zookeeper Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 12.1 Fixed and Floating Problems; Dilations . . . . . . . . . . . . . . . . . . . . . . . . 343 12.2 Solving the Safari Route Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 12.3 Solving the Zookeeper Route Problem . . . . . . . . . . . . . . . . . . . . . . . . . 351 12.4 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

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 less 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 B¨ulow 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 generalized 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 university level. It should also contain sufficient details for students and researchers in the field who are keen to understand the correctness proofs, the

7

8

Preface

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 optimizing 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 B¨ulow, 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 Kr¨uger, 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 Rosenˇ c for discussions or comfeld, the late Klaus Voss, Jinlong Wang, and Joviˇsa Zuni´ ments 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 Jos´e L. Marroqu´ın at CIMAT Guanajuato for an invitation to this institute, thus providing excellent conditions for working on this book project. Parts of Chapter 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 and Auckland, July 2011

Fajie Li Reinhard Klette

Part I

Discrete or Continuous Shortest Paths

The road system of the historic city of Guanajuato is mostly underground, forming a network of tunnels. Optimizing routes from one part of the city to another part is an exciting task not only for visitors of the city, but even for locals. In 2010, there was not yet a “3D route planner” available for calculating shortest connections in this historic city.

The first part of the book defines shortest paths in the geometry that we practice in our daily life, known as Euclidean geometry. Finding a shortest path between two given points and avoiding existing obstacles can be a challenge. We provide basic definitions and we propose the class of rubberband algorithms for solving various Euclidean shortest-path problems, either defined in the plane or in the 3-dimensional space.

Chapter 1

Euclidean Shortest Paths

Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by a study of the Elements, whereupon Euclid answered that there was no royal road to geometry. Proclus Diadochus (410...412 – 485)

This introductory chapter explains the difference between shortest paths in finite graphs and shortest paths in Euclidean geometry, which is also called ‘the common geometry of our world’. The chapter demonstrates the diversity of such problems, defined between points in a plane, on a surface, or in the 3-dimensional space.

1.1 Arithmetic Algorithms Technology is evaluated by applying various measures, which are mapping components of the technology into real numbers, such as kilometres per hour, a maximum error in millimetres, or a shape descriptor. We describe algorithms with respect to run time, deviations of results from being optimal, or necessary constraints on input data. The shortest path algorithms in this book are designed to be fast, accurate, and for a wide range of input data. The time for calculating a shortest path on a computer depends on parameters such as available memory space or execution time per applied operation. It is meaningless to express the required time as an absolute value in, say, micro- or nanoseconds, because computers differ in their parameters, they change frequently, and your computer is certainly not identical to one used by someone else. Measures for computing time need to be independent from the configuration of computers for expressing the quality of an algorithm. For example, we apply an abstract measure to estimate the running time of algorithms, also called time complexity or computational complexity. This is a common approach in computer science for more than 50 years. We define a set of basic operations, thus specifying a particular abstract computer, and we assign one time unit to each basic operation. Such a ‘unit of time’ is not measured in a physical scale; it is an abstract unit for each 11

12

1 Euclidean Shortest Paths

basic operation. For those abstract units, the estimation of run time of an algorithm is independent of progress in computer technology. Definition 1.1. An algorithm is a finite sequence of basic operations. We consider basic operations that are executable on ‘a normal sequential computer’, and not, for example, on some kind of specialized processor. Basic operations are classified into numerical operations, logical tests, and control instructions; alphanumerical operations on letters or other symbols are not a subject in this book. Logical tests are comparisons in magnitude (such as “length < threshold”?). Control instructions are of the type if-then-else, while-do, or do-until. It only remains to define a particular computer model by specifying the set of numerical operations. Definition 1.2. An arithmetic algorithm is defined by having addition, subtraction, multiplication, division, and the n-th root (for any integer n > 1) as its basic operations. A scientific algorithm expands an arithmetic algorithm by the following additional basic operations: trigonometric and inverse trigonometric functions, exponential and logarithmic functions, factorials, and conversions of numbers between different representation schemes (binary, decimal, or hexadecimal). This book applies arithmetic algorithms for solving shortest-path problems.

(There is just one case in the book where trigonometric functions are used when optimizing a point location in an ellipse.) Arithmetic algorithms are further specified by the range of numbers they are working on. For example, assuming the unbounded range of rational numbers a/b (for integers a and b) goes beyond the capabilities of existing computers, which can only represent rational numbers up to some finite number of bits per number. Even if divisions a/b are not performed, and assuming that there are no roots (i.e., all calculations remain in the area of integers) then there is still the problem of a limited range, having only a finite number of bits for representing all available integers. Definition 1.3. An arithmetic algorithm over the rational numbers uses the basic operations as available in an arithmetic algorithm and operates on the set of all rational numbers.

1.2 Upper Time Bounds We assume, as is common in algorithmic complexity theory, that the performance of each basic operation requires one time unit.