Maximal Palindromes in MPC: Simple and Optimal

TL;DR

This paper introduces a simple, optimal, and randomized MPC algorithm that computes the longest palindromic substring and all maximal palindromes in constant rounds with linear total time and memory, improving previous methods.

Contribution

It presents a new simple and optimal MPC algorithm for LPS that also computes all maximal palindromes and extends to the Adaptive MPC model without the previous memory constraints.

Findings

Achieves $ ext{O}(1)$ rounds for LPS in MPC.

Computes all maximal palindromes within the same complexity.

Successfully removes the $ ext{O}(n^{1- ext{epsilon}})$ memory constraint in the Adaptive MPC model.

Abstract

In the classical longest palindromic substring (LPS) problem, we are given a string $S$ of length $n$, and the task is to output a longest palindromic substring in $S$. Gilbert, Hajiaghayi, Saleh, and Seddighin [SPAA 2023] showed how to solve the LPS problem in the Massively Parallel Computation (MPC) model in $\mathcal{O}(1)$ rounds using $\mathcal{\widetilde{O}}(n)$ total memory, with $\mathcal{\widetilde{O}}(n^{1-\epsilon})$ memory per machine, for any $\epsilon \in (0,0.5]$. We present a simple and optimal algorithm to solve the LPS problem in the MPC model in $\mathcal{O}(1)$ rounds. The total time and memory are $\mathcal{O}(n)$, with $\mathcal{O}(n^{1-\epsilon})$ memory per machine, for any $\epsilon \in (0,0.5]$. A key attribute of our algorithm is its ability to compute all maximal palindromes in the same complexities. Furthermore, our new insights allow us to bypass the constraint $\epsilon \in (0,0.5]$ in the Adaptive MPC model. Our algorithms and the one proposed by Gilbert et al. for the LPS problem are randomized and succeed with high probability.