An improved algorithm for finding k-centrums on weighted trees

Location theory on networks has been widely investigated by researchers from different fields for more than thirty years due to its significance and practical value. Among various location problems, the p-center and the p-median problems are the most common. The p-facility k-centrum problem, introduced by Slater (1978), is a generalization of the above two problems. The objective is to minimize the sum of the k largest service distances from clients to their nearest servers. When p is an arbitrary integer, the problem is NP-hard on general networks. Therefore, most researchers have devoted to the single-facility case, i.e. p=1, or the case that the networks under consideration are trees. This paper focuses on the single-facility k-centrum problem on a tree. For this problem, Tamir (1996) had an O(nlog² n) time algorithm. In this paper, an O(nlog n) time algorithm with is proposed.