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

TL;DR

The paper addresses the problem of computing the exact Longest Increasing Subsequence (LIS) in the Massively Parallel Computation (MPC) model. It leverages the unit-Monge matrix framework to reduce LIS to implicit min-plus products, and then shows an $O(1)$-round, fully-scalable deterministic MPC algorithm for the implicit subunit-Monge product $P_A oxdot P_B$ when $P_A$ and $P_B$ are (sub-)permutation matrices. As a consequence, the LIS length can be computed in $O( obreak ext{log } n)$ rounds in fully scalable MPC, substantially improving the previous $O( obreak ext{log }^4 n)$ bound and matching the best $O(1)$-round approximate LIS results in more permissive models. The approach is deterministic, extends to the sub-permutation case, and has broader implications for semi-local LIS/LCS problems, establishing a new benchmark for exact parallel LIS computations in the MPC framework.

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 $(1+ε)$-approximation of LIS.

Theorems & Definitions (42)

• Theorem 1.1: Main Theorem
• Theorem 1.2
• Theorem 1.3
• Corollary 1.3.1
• Corollary 1.3.2
• Corollary 1.3.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: Inverse Permutation
• proof