An O(n log2 n) algorithm for the optimal sink location problem in dynamic tree networks

Authors:
Satoko Mamada;Takeaki Uno;Kazuhisa Makino;Satoru Fujishige
Affiliations:
Division of Mathematical Science for Social Systems, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan;Foundations of Informatics Research Division, National Institute of Informatics, Tokyo, Japan;Department of Mathematical Informatics, Graduate School of Information and Technology, University of Tokyo, Tokyo, Japan;Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
Venue:
Discrete Applied Mathematics - Special issue: Discrete algorithms and optimization, in honor of professor Toshihide Ibaraki at his retirement from Kyoto University
Year:
2006

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Abstract

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(nlog2n) 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 [S. Mamada, K. Makino, S. Fujishige, Optimal sink location problem for dynamic flows in a tree network, IEICE Trans. Fundamentals E85-A (2002) 1020-1025]. As a corollary, we also show that the quickest transshipment problem can be solved in O(nlog2n) 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.