An O(n log n)-Time Algorithm for the k-Center Problem in Trees

We consider a classical k-center problem in trees. Let T be a tree of n vertices and every vertex has a nonnegative weight. The problem is to find k centers on the edges of T such that the maximum weighted distance from all vertices to their closest centers is minimized. Megiddo and Tamir (SIAM J. Comput., 1983) gave an algorithm that can solve the problem in O(n log^2 n) time by using Cole\u27s parametric search. Since then it has been open for over three decades whether the problem can be solved in O(n log n) time. In this paper, we present an O(n log n) time algorithm for the problem and thus settle the open problem affirmatively

Geometric Facility Location Problems on Uncertain Data

Facility location, as an important topic in computer science and operations research, is concerned with placing facilities for serving demand points (each representing a customer) to minimize the (service) cost. In the real world, data is often associated with uncertainty because of measurement inaccuracy, sampling discrepancy, outdated data sources, resource limitation, etc. Hence, problems on uncertain data have attracted much attention. In this dissertation, we mainly study a classical facility location problem: the k- center problem and several of its variations, on uncertain points each of which has multiple locations that follow a probability density function (pdf). We develop efficient algorithms for solving these problems. Since these problems more or less have certain geometric flavor, computational geometry techniques are utilized to help develop the algorithms. In particular, we first study the k-center problem on uncertain points on a line, which is aimed to find k centers on the line to minimize the maximum expected distance from all uncertain points to their expected closest centers. We develop efficient algorithms for both the continuous case where the location of every uncertain point follows a continuous piecewise-uniform pdf and the discrete case where each uncertain point has multiple discrete locations each associated with a probability. The time complexities of our algorithms are nearly linear and match those for the same problem on deterministic points. Then, we consider the one-center problem (i.e., k= 1) on a tree, where each uncertain point has multiple locations in the tree and we want to compute a center in the tree to minimize the maximum expected distance from it to all uncertain points. We solve the problem in linear time by proposing a new algorithmic scheme, called the refined prune-and-search. Next, we consider the one-dimensional one-center problem of uncertain points with continuous pdfs, and the one-center problem in the plane under the rectilinear metric for uncertain points with discrete locations. We solve both problems in linear time, again by using the refined prune-and-search technique. In addition, we study the k-center problem on uncertain points in a tree. We present an efficient algorithm for the problem by proposing a new tree decomposition and developing several data structures. The tree decomposition and these data structures may be interesting in their own right. Finally, we consider the line-constrained k-center problem on deterministic points in the plane where the centers are required to be located on a given line. Several distance metrics including L1, L2, and L1 are considered. We also study the line-constrained k-median and k-means problems in the plane. These problems have been studied before. Based on geometric observations, we design new algorithms that improve the previous work. The algorithms and techniques we developed in this dissertation may and other applications as well, in particular, on solving other related problems on uncertain data