Abstract
Consider an unweighted, directed graph G with the diameter D. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time widetilde {O}(n^{omega }). The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the All-Nodes Shortest Cycles, All-Pairs All Walks problems efficiently and also give some improvement upon distance queries in unweighted graphs.
Highlights
The All-Pairs Shortest Paths (APSP) problem asks to find distances between all pairs of vertices in a graph
For a given graph G and k determining whether G contains a simple cycle of length exactly k is NP-hard
We introduced the framework based on Frobenius normal form and used it to solve some problems on directed, unweighted graphs in matrix multiplication time
Summary
The All-Pairs Shortest Paths (APSP) problem asks to find distances between all pairs of vertices in a graph. After a long line of improvements [9] showed an O(Mnω) time algorithms for finding minimum weight perfect matching, shortest cycle, diameter and radius (some of these results were already known [21]). [9] showed an application of their techniques that improves upon [31] O(Mnωt) time algorithm for the following problem: determine the set of vertices that lie on some cycle of length at most t. Cygan et al [9] managed to solve this problem in O(Mnω) time using Baur-Strassen’s theorem All of these algorithm are effective merely in the case of a dense graphs. For graphs with the small number of edges there are more efficient algorithms (e.g., APSP in O(|V ||E|) time [27]).
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