Core-Sparse Monge Matrix Multiplication: Improved Algorithm and Applications

Abstract

The task of min-plus matrix multiplication often arises in the context of distances in graphs and is known to be fine-grained equivalent to the All-Pairs Shortest Path problem. The non-crossing property of shortest paths in planar graphs gives rise to Monge matrices; the min-plus product of $n\times n$ Monge matrices can be computed in $O(n^2)$ time. Grid graphs arising in sequence alignment problems, such as longest common subsequence or longest increasing subsequence, are even more structured. Tiskin [SODA'10] modeled their behavior using simple unit-Monge matrices and showed that the min-plus product of such matrices can be computed in $O(n\log n)$ time. Russo [SPIRE'11] showed that the min-plus product of arbitrary Monge matrices can be computed in time $O((n+\delta)\log^3 n)$ parameterized by the core size $\delta$, which is $O(n)$ for unit-Monge matrices. In this work, we provide a linear bound on the core size of the product matrix in terms of the core sizes of the input matrices and show how to solve the core-sparse Monge matrix multiplication problem in $O((n+\delta)\log n)$ time, matching the result of Tiskin for simple unit-Monge matrices. Our algorithm also allows $O(\log \delta)$-time witness recovery for any given entry of the output matrix. As an application of this functionality, we show that an array of size $n$ can be preprocessed in $O(n\log^3 n)$ time so that the longest increasing subsequence of any sub-array can be reconstructed in $O(l)$ time, where $l$ is the length of the reported subsequence; in comparison, Karthik C. S. and Rahul [arXiv'24] recently achieved $O(l+n^{1/2}\log^3 n)$-time reporting after $O(n^{3/2}\log^3 n)$-time preprocessing. Our faster core-sparse Monge matrix multiplication also enabled reducing two logarithmic factors in the running times of the recent algorithms for edit distance with integer weights [Gorbachev & Kociumaka, arXiv'24].