Faster Algorithms for Bounded-Difference Min-Plus Product

10/17/2021

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by   Shucheng Chi, et al.

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Min-plus product of two n× 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 δ-bounded-difference matrices, in which the difference between any two adjacent entries is bounded by δ=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 Õ(n^2+ω/3)=O(n^2.791) (ω<2.373 [Alman & V.V.Williams 20]). This improves previous result of Õ(n^2.824) [Bringmann et al. 16]. When ω=2 in the ideal case, our complexity is Õ(n^2+2/3), improving Bringmann et al.'s result of Õ(n^2.755).