Improved Distance Queries and Cycle Counting by Frobenius Normal Form

TL;DR

This paper presents a new framework using Frobenius normal form and Hankel matrix techniques to efficiently count cycles and walks, and improve distance queries in unweighted directed graphs.

Contribution

It introduces a novel approach combining Frobenius normal form and Hankel matrices for faster cycle counting and distance querying in directed graphs.

Findings

Cycle and walk counting achieved in matrix multiplication time $ ilde{O}(n^)$.

Improved algorithms for All-Nodes Shortest Cycles and All-Pairs All Walks.

Enhanced distance query performance in unweighted directed graphs.

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.