The Online Median Problem

We introduce a natural variant of the (metric uncapacitated) k-median problem that we call the online median problem. Whereas the k-median problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities are placed one at a time, a facility cannot be moved once it is placed, and the total number of facilities to be placed, k, is not known in advance. The objective of an online median algorithm is to minimize the competitive ratio, that is, the worst-case ratio of the cost of an online placement to that of an optimal offline placement. Our main result is a constant-competitive algorithm for the online median problem running in time that is linear in the input size. In addition, we present a related, though substantially simpler, constant-factor approximation algorithm for the (metric uncapacitated) facility location problem that runs in time linear in the input size. The latter algorithm is similar in spirit to the recent primal-dual-based facility location algorithm of Jain and Vazirani, but our approach is more elementary and yields an improved running time. While our primary focus is on problems which ask us to minimize the weighted average service distance to facilities, we also show that our results can be generalized to hold, to within constant factors, for more general objective functions. For example, we show that all of our approximation results hold, to within constant factors, for the k-means objective function

Approximation Algorithms for Hierarchical Location Problems

We formulate and (approximately) solve hierarchical versions of two prototypical problems in discrete location theory, namely, the metric uncapacitated k-median and facility location problems. Our work yields new insights into hierarchical clustering, a widely used technique in data analysis. First, we show that every metric space admits a hierarchical clustering that is within a constant factor of optimal at every level of granularity with respect to the average (squared) distance objective. Second, we provide a natural solution to the leaf ordering problem encountered in the traditional dendrogram-based approach to the visualization of hierarchical clusterings

Optimal Cover Time for a Graph-Based Coupon Collector Process

Journal of Discrete Algorithms, February, 2013The article of record as published may be located at http://dx.doi.org/10.1016/j.jda.2013.02.003In this paper we study the following covering process defined over an arbitrary directed graph. Each node is initially uncovered and is assigned a random integer rank drawn from a suitable range. The process then proceeds in rounds. In each round, a uniformly random node is selected and its lowest-ranked uncovered outgoing neighbor, if such exists, is covered..

A Hypercubic Sorting Network with Nearly Logarithmic Depth

A natural class of "hypercubic" sorting networks is defined. The regular structure of these sorting networks allows for elegant and efficient implementations on any of the so-called hypercubic networks (e.g., the hypercube, shuffle-exchange, butterfly, and cube-connected cycles). This class of sorting networks contains Batcher's O(lg 2 n)-depth bitonic sort, but not the O(lg n)-depth sorting network of Ajtai, Koml'os, and Szemer'edi. In fact, no o(lg 2 n)- depth compare-interchange sort was previously known for any of the hypercubic networks. In this paper, we prove the existence of a family of 2 O( p lg lg n) lg n-depth hypercubic sorting networks. Note that this depth is o(lg 1+ffl n) for any constant ffl ? 0. 1 Introduction A comparator network is an n-input, n-output acyclic circuit made up of wires and 2-input, 2output comparator gates. The input wires of the network are numbered from 0 to n \Gamma 1, as are the output wires. The inputs to the network may be tho..

Approximation Algorithms for Hierarchical Location Problems (Extended Abstract)

C. Greg Plaxton Department of Computer Science University of Texas at Austin [email protected] ABSTRACT We formulate and (approximately) solve hierarchical versions of two prototypical problems in discrete location theory, namely, the metric uncapacitated k-median and facility location problems. Our work yields new insights into hierarchical clustering, a widely used technique in data analysis. First, we show that every metric space admits a hierarchical clustering that is within a constant factor of optimal at every level of granularity with respect to the average (squared) distance objective. Second, we provide a natural solution to the leaf ordering problem encountered in the traditional dendrogram-based approach to the visualization of hierarchical clusterings