Truly Subcubic Min-Plus Product for Less Structured Matrices, with Applications

Abstract

The goal of this paper is to get truly subcubic algorithms for Min-Plus product for less structured inputs than what was previously known, and to apply them to versions of All-Pairs Shortest Paths (APSP) and other problems. The results are as follows: (1) Our main result is the first truly subcubic algorithm for the Min-Plus product of two $n\times n$ matrices $A$ and $B$ with $\text{polylog}(n)$ bit integer entries, where $B$ has a partitioning into $n^{\epsilon}\times n^{\epsilon}$ blocks (for any $\epsilon>0$) where each block is at most $n^\delta$-far (for $\delta<3-\omega$, where $2\leq \omega<2.373$) in $\ell_\infty$ norm from a constant rank integer matrix. This result presents the most general case to date of Min-Plus product that is solvable in truly subcubic time. (2) The first application of our main result is a truly subcubic algorithm for APSP in a new type of geometric graph. Our result extends the result of Chan'10 in the case of integer edge weights by allowing the weights to differ from a function of the end-point identities by at most $n^\delta$ for small $\delta$. (3) In the second application we consider a batch version of the range mode problem in which one is given a length $n$ sequence and $n$ contiguous subsequences, and one is asked to compute the range mode of each subsequence. We give the first $O(n^{1.5-\epsilon})$ time for $\epsilon>0$ algorithm for this batch range mode problem. (4) Our final application is to the Maximum Subarray problem: given an $n\times n$ integer matrix, find the contiguous subarray of maximum entry sum. We show that Maximum Subarray can be solved in truly subcubic, $O(n^{3-\epsilon})$ (for $\epsilon>0$) time, as long as the entries are no larger than $O(n^{0.62})$ in absolute value. We also improve all the known conditional hardness results for the $d$-dimensional variant of Maximum Subarray.