An O(n log 2n) Algorithm for the Optimal Sink Location Problem in Dynamic Tree Networks

In this paper, we consider a sink location in a dynamic network which consists of a graph with capacities and transit times on its arcs. Given a dynamic network with initial supplies at vertices, the problem is to find a vertex v as a sink in the network such that we can send all the initial supplies to v as quickly as possible. We present an O(n log n) time algorithm for the sink location problem, in a dynamic network of tree structure where n is the number of vertices in the network. This improves upon the existing O(n2)-time bound. As a corollary, we also show that the quickest transshipment problem can be solved in O(n log n) time if a given network is a tree and has a single sink. Our results are based on data structures for representing tables (i.e., sets of intervals with their height), which may be of independent interest.