Efficient Algorithms for Two Generalized 2-Median Problems on Trees

The p-median problem on a tree T is to find a set S of p vertices on T that minimize the sum of distances from T's vertices to S. For this problem, Tamir [12] had an O(pn 2)-time algorithm, while Gavish and Sridhar [1] had an O(nlog n)-time algorithm for the case of p=2. Wang et al. [13] introduced two generalizations by imposing constraints on the 2-median: one is to limit their distance while the other is to limit their eccentricity, and they had O(n2)-time algorithms for both. We solve both generalizations in O(nlog n) time, matching even the fastest algorithm currently known for the 2-median problem. We also study cases when linear time algorithms exist for the 2-median problem and the two generalizations. For example, we solve all three in linear time when edge lengths and vertex weights are all polynomially bounded integers. Finally, 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(nlog n)-time algorithms for them.