Efficient algorithms for minimum range cut problems

Let G = (V, E) be an undirected graph with n vertices and m edges such that a real‐valued weight, denoted by w(e), is associated with each edge e. This paper studies what we call the minimum range cut problem that asks to find a cut in G such that the range of all edge weights in the cut is minimum. Here, the range of a cut C is defined to be the maximum difference among weights of edges in the cut, i.e., maxeϵc w(e) ‐ mineϵc w(e). This paper proposes an O(m + n log n) algorithm for the minimum range cut problem. It is also shown that this running time is optimal. We also study two variants of this problem. One is the minimum range target cut problem. Given a prespecified value called a target, this problem asks to find a cut with minimum range among all cuts such that the target value is between the minimum and maximum of weights of edges in the cut. The second is the minimum range s – t cut problem that asks to find an s – t cut with minimum range. This paper proposes O(m + n log n) algorithms for these problems. For the second problem, we show that an ancestor tree of O(n) space recently developed by Cheng and Hu effectively represents all pairs minimum range cuts, which can be constructed in O(n2) time, and enables us to answer any minimum range s – t cut query in O(1) time [resp., in O(n) time] if we want to obtain only the range value of the cut (resp., the bipartition of vertices induced by the cut). © 1994 by John Wiley & Sons, Inc.