Improved Distance Queries and Cycle Counting by Frobenius Normal Form

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 $\tilde{O}(n^{\omega })$. 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.

Lemma 2 in the previous version of this paper is incorrect. We thank Adam Karczmarz for pointing out the mistake

References in corpus (1)

• Powers of Tensors and Fast Matrix Multiplication