Efficient algorithms for two generalized 2-median problems and the group median problem on trees

The p-median problem on a tree T is to find a set S of p vertices on T that minimizes the sum of distances from T’s vertices to S. In this paper, we study two generalizations of the 2-median problem, which are obtained by imposing constraints on the two vertices selected as a 2-median: one is to limit their distance while the other is to limit their eccentricity. Previously, both the best upper bounds of these two generalizations were O(n2) [A. Tamir, D. Perez-Brito, J.A. Moreno-Perez, A polynomial algorithm for the p-centdian problem on a tree, Networks 32 (1998) 255–262; B.-F. Wang, S.-C. Ku, K.-H. Shi, Cost-optimal parallel algorithms for the tree bisector problem and applications, IEEE Transactions on Parallel and Distributed Systems 12 (9) (2001) 888–898]. In this paper, we solve both in O(nlogn) time. We also study cases when linear time algorithms exist for the two generalizations. For example, we solve both in linear time when edge lengths and vertex weights are all polynomially bounded integers. Furthermore, we consider the relaxation of the two generalized problems by allowing 2-medians on any position of edges, instead of just on vertices, and we give O(nlogn)-time algorithms for them. A problem, named the tree marker problem, arises several times in our approaches to the two generalized 2-median problems, and we give an O(nlogn)-time algorithm for this problem. We also use this algorithm to speedup an algorithm of Gupta and Punnen [S.K. Gupta, A.P. Punnen, Group center and group median of a tree, European Journal of Operational Research 65 (1993) 400–406] for the group median problem, improving the running time from O(kn) to O(n+klogn), where k is the number of groups in the input.