Frontier Space-Time Algorithms Using Only Full Memory

TL;DR

This paper introduces catalytic algorithms that operate in polynomial time with minimal workspace, matching the best known bounds of non-catalytic algorithms for fundamental problems like graph connectivity and sequence comparison.

Contribution

It presents the first polynomial time catalytic algorithms for key problems that match non-catalytic time-space bounds, using only logarithmic workspace and random bits.

Findings

Directed s-t connectivity algorithm with $n/2^{ heta( oot{ ext{log n}})}$ catalytic space.

Algorithms for Edit Distance, LCS, and Fréchet Distance with similar catalytic space bounds.

Matching non-catalytic time-space bounds for multiple fundamental problems.

Abstract

We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only $\mathcal{O}(\log(n))$ workspace, and use sublinear catalytic space matching the best-known space bounds of non-catalytic algorithms running in polynomial time. First, we design a polynomial time algorithm for directed $s$-$t$ connectivity using $n \big/ 2^{\Theta(\sqrt{\log n})}$ catalytic space, which matches the state-of-the-art time-space bounds in the non-catalytic setting [Barnes et al., 1998], and improves the catalytic space usage of the best known algorithm [Cook and Pyne, 2026]. Furthermore, using only $\mathcal{O}(\log(n))$ random bits we get a randomized algorithm whose running time nearly matches the fastest time bounds known for space-unrestricted algorithms. Second, we design polynomial time algorithms for the problems of computing Edit Distance, Longest Common Subsequence, and the Discrete Fr\'{e}chet Distance, again using $n \big/ 2^{\Theta(\sqrt{\log n})}$ catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].