An Optimal MPC Algorithm for Subunit-Monge Matrix Multiplication, with Applications to LIS

TL;DR

This paper introduces an efficient parallel algorithm for min-plus matrix multiplication of unit-Monge matrices and applies it to significantly improve the parallel complexity of solving the LIS problem.

Contribution

It presents the first $O(1)$-round deterministic MPC algorithm for unit-Monge matrix multiplication and a faster $O( ext{log} n)$-round algorithm for LIS, surpassing previous methods.

Findings

Achieves $O(1)$-round MPC for unit-Monge matrix multiplication.

Develops $O( ext{log} n)$-round MPC for exact LIS.

Improves previous LIS algorithms from $O( ext{log}^4 n)$ to $O( ext{log} n)$).

Abstract

We present an $O(1)$-round fully-scalable deterministic massively parallel algorithm for computing the min-plus matrix multiplication of unit-Monge matrices. We use this to derive a $O(\log n)$-round fully-scalable massively parallel algorithm for solving the exact longest increasing subsequence (LIS) problem. For a fully-scalable MPC regime, this result substantially improves the previously known algorithm of $O(\log^4 n)$-round complexity, and matches the best algorithm for computing the $(1+\epsilon)$-approximation of LIS.