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Path Planning Algorithms Under the Link-Distance Metric Dartmouth College Dartmouth Digital Commons Dartmouth College Ph.D Dissertations Theses and Dissertations 2-1-2006 Path Planning Algorithms under the Link-Distance Metric David Phillip Wagner Dartmouth College Follow this and additional works at: https://digitalcommons.dartmouth.edu/dissertations Part of the Computer Sciences Commons Recommended Citation Wagner, David Phillip, "Path Planning Algorithms under the Link-Distance Metric" (2006). Dartmouth College Ph.D Dissertations. 16. https://digitalcommons.dartmouth.edu/dissertations/16 This Thesis (Ph.D.) is brought to you for free and open access by the Theses and Dissertations at Dartmouth Digital Commons. It has been accepted for inclusion in Dartmouth College Ph.D Dissertations by an authorized administrator of Dartmouth Digital Commons. For more information, please contact [email protected]. Path Planning Algorithms under the Link-Distance Metric A Thesis Submitted to the Faculty in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science by David Phillip Wagner DARTMOUTH COLLEGE Hanover, New Hampshire February, 2006 Examining Committee: (co-chair) Robert L. (Scot) Drysdale III (co-chair) Clifford Stein Amit Chakrabarti Joseph S. B. Mitchell Charles K. Barlowe Dean of Graduate Studies c Copyright by David Phillip Wagner 2006 Abstract of the Dissertation Path Planning Algorithms under the Link-Distance Metric by David Phillip Wagner Doctor of Philosophy in Computer Science Dartmouth College, Hanover, NH February 2006 Professor Robert Scot Drysdale, Co-Chair Professor Clifford Stein, Co-Chair The Traveling Salesman Problem and the Shortest Path Problem are famous problems in computer science which have been well studied when the objective is measured using the Euclidean distance. Here we examine these geometric problems under a different set of optimization criteria. Rather than considering the total distance traversed by a path, this thesis looks at reducing the number of times a turn is made along that path, or equivalently, at reducing the number of straight lines in the path. Minimizing this objective value, known as the link-distance, is useful in situations where continuing in a given direction is cheap, while turning is a relatively expensive op- eration. Applications exist in VLSI, robotics, wireless communications, space travel, and other fields where it is desirable to reduce the number of turns. This thesis examines rectilinear and non-rectilinear variants of the Traveling Salesman Problem under this metric. The objective of these problems is to find a path visiting a set of points which has the smallest number of bends. A 2-approximation algorithm is given for the rectilinear problem, while for the non-rectilinear problem, an O(log n)-approximation algorithm is given. The latter problem is also shown to be NP-Complete. ii Next, the Rectilinear Minimum Link-Distance Problem, also known as the Minimum Bends Path Problem, is considered. Here the objective is to find a rectilinear path between two points among rectilinear obstacles which has the minimum number of bends, while avoiding passing through any of the obstacles. The problem has been well studied in two dimensions, but is relatively unexplored in higher dimensions. A main result of this thesis is an O(n5=2 log n) time algorithm solving this problem in three dimensions. Previously known algorithms have had worst-case running times of Ω(n3). This algorithm requires a data structure that supports efficient operations on pointsets within rectangular regions of the Euclidean plane. We design a new data structure, which is a variation on the segment tree, in order to support these operations. Finally, an implementation of the data structure and of the algorithm solving the Min- imum Link-Distance Problem demonstrates their experimental running times and ease of implementation. iii Biographical Sketch David Phillip Wagner was born in Aberdeen, Maryland on February 5, 1971. At the age of three his family moved to Bloomfield, Connecticut where he lived until the age of eighteen. In August 1989 he left home to attend Colgate University in Hamilton, NY. At Colgate he earned a Bachelor of Arts degree with a dual concentration in Computer Science and Mathematics, working under advisors Phil Mulry in Computer Science and Tom Tucker in Mathematics, in May 1993. From there he went on to pursue a Master of Science degree in Computer Science at the State University of New York at Stony Brook, working under advisor Steven Skiena, and graduating in May 1995. After finishing his Master's degree he worked for three years in New York City. The first six months of these were spent working at Wanderlust Interactive in SoHo, and the remainder was spent working downtown in the financial district for JYACC/Prolifics. In September 1998 he returned to the academia to pursue a Doctorate in Computer Science at Dartmouth College, under co-advisors Clifford Stein and Robert Scot Drysdale. It is perhaps interesting to note that although Colgate calls itself a liberal arts college, it retains the title of “University” due to a small graduate program which awards about ten Master's degrees each year. And, although Dartmouth has an active graduate program including schools of business, medicine, engineering, and dozens of fields in the arts and sciences, it retains the title of “College” due to the outcome of an 1818 Supreme Court case where a state run Dartmouth University was prevented from replacing Dartmouth College. iv Acknowledgments I would like to extend my heartfelt thanks to my thesis committee for their advice and the time they have spent considering this dissertation, and for the many fruitful discussions they have provided. I would especially like to thank my co-advisor Robert Scot Drysdale whose fastidious attention to the details of my work have proven invaluable, especially in Chapters 4 and 5, and my co-advisor Clifford Stein who inspired me to pursue this line of research in a moment of creativity and who advised me throughout the writing of Chapter 3. I must also thank Neal Young, who guided me through the very earliest incarnations of this research, especially that appearing in Sections 3.3 and 3.4.1. Along the way, many talented researchers have given their time and advice to further the results in this thesis. These include Michel X. Goemans (Section 3.4.1), Takeshi Tokuyama (Section 4.4.2), David R. Karger (Chapter 2), Jon L. Bentley (Section 7.2.1), Gunter¨ Rote (Section 7.2.1), Tatsuo Ohtsuki (Section 1.1), Joseph S. B. Mitchell (Sections 1.1, 3.6.5, and 7.2.4), Valentin Polishchuk (Section 1.1), and Robert Fitch (Chapters 2 and 4). I am grateful for their advice which illuminated many deficiencies, and allowed many further results within this work. I am deeply indebted to numerous faculty who have encouraged, enlightened, and in- spired me through the years, and who have taught me to appreciate a lifelong pursuit of learning. I would especially thank Prasad Jayanti, Javed Aslam, Amit Chakrabarti, and Kenneth Bogart from Dartmouth, Steven Skiena and Joseph Mitchell from SUNY Stony v Brook, and Laura Sanchis, Phil Mulry, Tom Tucker, and Dan Saracino from Colgate. It would not be an exaggeration to say that every one of these people has substantially changed my life for the better. I am very grateful to the NSF Summer Institute Program, for their sponsorship of re- search summers at IBM Tokyo Research Laboratory in Yamato, Japan, and at Korea Ad- vanced Institute of Science and Technology (KAIST) in Taejon, Korea. The memories I have from these trips will undoubtedly last a lifetime. I consider myself incredibly fortunate to have met many researchers abroad who were kind enough to invite me into their labo- ratories in order to broaden my understanding of how research is done, including Takao Nishizeki from Tohoku University, Takeshi Tokuyama from Tohoku University, Satoru Iwata from Tokyo University, Tetsuo Asano from JAIST, Kyung-Yong Chwa from KAIST, Kyunsoo Park from Seoul National Universtiy, Mordecai Golin from HKUST, Tak-Wah Lam from Hong Kong Universtiy, and Der-Tsai Lee from Academia Sincia. Additionally, the faculty of the Department of Japanese at Dartmouth and at Stony Brook have been very gracious in repeatedly permitting me to audit their classes. I am grateful especially to Mayumi Ishida, Ikuko Watanabe, Dennis Washburn, and James Dorsey. Without their help I surely would have forgotten a fascinating and beautiful language. Many clubs and organizations I have participated in while at Dartmouth have made my time here rich and enjoyable. I am fortunate to have been involved with the Handel So- ciety, the Dartmouth Japan Society, the Dartmouth Argentine Tango Society, Pound Bang Slash Intramural Hockey Team, Korean Miniversity, UFX Ultimate Frisbee, MEaD, and the Dartmouth Bridge Club, among others. Thanks to Bhavnesh at the India Queen, Ken and the uvScene, the Tabard, the Hop, and the Lebanon Opera House for providing many ways to warm up those cold New Hampshire nights. My friends at Dartmouth have undoubtedly made this a more memorable place. Ed, Chilann, Marina, You-Shan, Luma, my two years in North Park were some of the best of vi my life. LeeAnn, Ming, Kazuhiro, Zhengyi, this department has been a richer place thanks to you. Guanling, Tony, Clint, Mark, Lea, Song, I'm fortunate to have had such wonderful officemates. Ji Yeon, James, Kim, thanks for many helpful language lessons. George, Sue Mei, Helen, and all of DJS, I hope we can meet at another karaoke bar someday. Wendy, Ashley, Yulie, I'll miss seeing you at those medieval feasts. Liya, Katie, Brian, Barry, Karen, Daisy, good luck with the Argentine Tango club. I especially want to thank some friends I've made outside of Dartmouth. Robyn, Lau- ren, and everyone at the Center for Cartoon Studies in White River Junction, I hope you and your fledgling school will be a great success.
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