New Results on the Complexity of p-Center Problems

An O(n log3 n) algorithm for the continuous p-center problem on a tree is presented. Following a sequence of previous algorithms, ours is the first one whose time bound in uniform in p and less than quadratic in n. We also present an O(n log2 n log log n) algorithm for a weighted discrete p-center problem. 1. Introduction. The p-center problems are defined on a weighted undirected graph G = (V, E). Each edge (i, j) has a positive length dip An edge is identified with a line segment of length dij so that we can refer to any "point" on (i, j) at a distance of t from i and dij-t from j (0 5 t 5 dij). The set of all such points of the graph is denoted, by A. If x, y E A then by d (x , y) we mean the length of the shortest path from x to y.