Subcubic Min-Plus Product of Structured Matrices

Abstract

The All-Pairs Shortest Paths (APSP) problem is one of the most basic problems in computer science. The fastest known algorithms for APSP in 𝑛-node graphs run in 𝑛³⁻⁰⁽¹⁾ time, and it is a big open problem whether a truly subcubic, 𝑂(𝑛³⁻ superscript 𝜀) for 𝜀 > 0 time algorithm exists for APSP. The Min-Plus product of two 𝑛 × 𝑛 matrices is known to be equivalent to APSP, where the optimal running times of the two problems differ by at most a constant factor. A natural way to approach understanding the complexity of APSP is thus understanding what structure (if any) is needed to solve Min-Plus Product in truly subcubic time. The goal of this thesis is to get truly subcubic algorithms for Min-Plus products for less structured inputs than what was previously known, and to apply them to versions of APSP and other problems. This thesis gives sub-cubic algorithms for two interesting cases of structured Min-Plus Products: Min-Plus product between matrices with a constant additive approximate rank and Min-Plus product between monotone matrices, whose definitions are deferred to the main text. These faster algorithms have a wide range of applications, including Geometric APSP, Maximum Subarray, Range Mode and Single Source Replacement Paths.