Minimax and Maximin Fitting of Geometric Objects to Sets of Points (original) (raw)

This thesis addresses several problems in the facility location sub-area of computational geometry. Let S be a set of n points in the plane. We derive algorithms for approximating S by a step function curve of size k < n, i.e., by an x-monotone orthogonal polyline R with k < n horizontal segments. We use the vertical distance to measure the quality of the approximation, i.e., the maximum distance from a point in S to the horizontal segment directly above or below it. We consider two types of problems: min-ε, where the goal is to minimize the error for a given number of horizontal segments k and min-#, where the goal is to minimize the number of segments for a given allowed error ε. After O(n) preprocessing time, we solve instances of the latter in O(min{k log n, n}) time per instance. We can then solve the former problem in O(min{n 2 , nk log n}) time. Both algorithms require O(n) space. The second contribution is a heuristic for the min-ε problem that computes a solution within a factor of 3 of the optimal error for k segments, or with at most the same error as the k-optimal but using 2k − 1 segments. Furthermore, experiments on real data show even better results than what is guaranteed by the theoretical bounds. Both approximations run in O(n log n) time and O(n) space. Then, we present an exact algorithm for the weighted version of this problem that runs in O(n 2) time and generalize the heuristic to handle weights at the expense of an additional log n factor. At this point, a randomized I thank my advisor Mario Lopez for his many years of close friendship and mentoring. His patience, advice, and consistent encouragement were absolutely invaluable to me in completing this dissertation and degree. I also thank my committee members, Alvaro Arias, Chris GauthierDickey, Scott Leutenegger, and Ronald DeLyser, for their time, advice, and comments. In addition, I would like to thank Drs. GauthierDickey and DeLyser for the many grammatical and presentation suggestions to my thesis. I thank again my advisor Mario Lopez for inviting me to continue my graduate school experience with doctoral work. I also want to thank the faculty at the departments of Computer Science and Mathematics whose courses I have taken at the University of Denver throughout all my degrees. I am grateful to all of you. Last, but not least, I wholeheartedly thank my graduate student colleagues and coauthors, Mohammed Al-Bow and Riquelmi Cardona, for their friendship, collaboration, conversations, and all the time we have spent together. I also thank Jeff Edgington for leading the way and showing me the path to graduation and freedom. All of you friends and colleagues, whose names are too many to mention here, have imparted your knowledge and wisdom to me, and I thank you all. Most importantly, I thank my parents, Boris and Rita Mayster, and my brother, Dmitriy Mayster, for their love and attention.