Some Insights on Dynamic Maintenance of Gomory-Hu Tree in Cactus Graphs and General Graphs

Abstract

AbstractFor any flow network, min(s, t)-cut query is a fundamental graph query that asks for a minimum weight cut that separates vertices s and t. Gomory and Hu [13] proposed a data structure which is an undirected weighted tree that compactly stores min(s, t)-cut for all (s, t) pairs of an undirected weighted graph. Although there have been some research towards the problem of dynamically maintaining Gomory-Hu tree of a graph [4, 18], an efficient dynamic (incremental or decremental) algorithm for general graphs remains elusive. Also efficient dynamic algorithms for maintenance of Gomory-Hu tree for special graphs has not been investigated sufficiently. In this paper we propose algorithms for Gomory-Hu tree for a special class of graphs, known as cactus graphs. First we show that Gomory-Hu tree for a cactus graph can be constructed in linear time. Then we provide both incremental and decremental algorithms for maintaining a Gomory-Hu tree of a cactus graph. The algorithms use relations between blocks of a graph and its Gomory-Hu tree. For the incremental algorithm the amortized update time is \(O(\log n)\) and for the decremental algorithm the worst-case update time is \(O(\log n)\). For general graphs with integral weights, we present a data structure requiring \(O(mn^2)\) space that helps us create a new Gomory-Hu tree if the weights of some edges of a given graph are changed by some integral amounts. Specifically, if the weights of k edges are changed by \(w_{1}, w_{2}, ..., w_{k}\) units respectively, then a new Gomory-Hu tree of the modified graph can be reconstructed in \(O((\sum _{i=1}^{k}w_{i})mn)\) time.KeywordsMin-cutGomory-Hu treeCactus graphDynamic graph