Computer Science – Data Structures and Algorithms
Scientific paper
2012-04-07
Computer Science
Data Structures and Algorithms
Scientific paper
Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: - Finding Shortest Cycles - We give a simple O(Wn^\omega) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs this matches the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the other hand, no algorithm working in $\tilde{O}(Wn^{\omega})$ time was previously known for undirected graphs with negative weights. - Computing Diameter - We give a simple O(Wn^\omega) time algorithm for computing a diameter of an undirected or directed graphs. This considerably improves the bounds of Yuster from 2010, who was able to obtain this time bound only in the case of directed graphs with positive weights. In contrary, our algorithm works in the same time bound for both directed and undirected graphs with negative weights. - Finding Minimum Weight Perfect Matchings - We present an O(Wn^\omega) time algorithm for finding minimum weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski in 2006, who presented such an algorithm but only in the case of bipartite graphs. We believe that the presented framework can find applications for solving larger spectra of related problems. As an illustrative example we apply it to the problem of computing a set of vertices that lie on cycles of length at most t, for some t. We give a simple O(Wn^\omega) time algorithm for this problem that improves over the O(tWn^\omega}) time algorithm given by Yuster in 2011.
Cygan Marek
Gabow Harold N.
Sankowski Piotr
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