Faster Algorithms for Bounded-Difference Min-Plus Product

TL;DR

This paper presents a faster randomized algorithm for min-plus matrix multiplication on $ ext{delta}$-bounded-difference matrices, improving the complexity over previous methods and leveraging advanced matrix multiplication techniques.

Contribution

Introduces a new randomized algorithm for min-plus product on bounded-difference matrices with improved runtime using fast rectangular matrix multiplication.

Findings

Achieves $O(n^{2.779})$ runtime, better than previous $O(n^{2.824})$.

Improves theoretical bounds for special matrix classes.

Leverages advanced matrix multiplication algorithms for efficiency.

Abstract

Min-plus product of two $n\times n$ matrices is a fundamental problem in algorithm research. It is known to be equivalent to APSP, and in general it has no truly subcubic algorithms. In this paper, we focus on the min-plus product on a special class of matrices, called $\delta$-bounded-difference matrices, in which the difference between any two adjacent entries is bounded by $\delta=O(1)$. Our algorithm runs in randomized time $O(n^{2.779})$ by the fast rectangular matrix multiplication algorithm [Le Gall \& Urrutia 18], better than $\tilde{O}(n^{2+\omega/3})=O(n^{2.791})$ ($\omega<2.373$ [Alman \& V.V.Williams 20]). This improves previous result of $\tilde{O}(n^{2.824})$ [Bringmann et al. 16]. When $\omega=2$ in the ideal case, our complexity is $\tilde{O}(n^{2+2/3})$, improving Bringmann et al.'s result of $\tilde{O}(n^{2.755})$.