Frontier Space-Time Algorithms Using Only Full Memory

2026-02-24

AI Summary

This work investigates the efficient solution of classic computational problems—including directed s-t connectivity, edit distance, longest common subsequence, and discrete Fréchet distance—under the stringent constraint of using only O(log n) working space. By leveraging the catalytic computation model, the study designs polynomial-time algorithms that utilize a read-only catalytic space of size n / 2^{Θ(√log n)} to achieve, within O(log n) working space, time-space trade-offs approaching those of optimal non-catalytic algorithms. The key contribution lies in the first matching of state-of-the-art time-space bounds of non-catalytic algorithms within the catalytic framework, while simultaneously reducing the number of random bits required by randomized algorithms to O(log n), thereby attaining near-optimal runtime efficiency comparable to settings without space restrictions.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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].

Problem

• catalytic space
• space-time tradeoff
• s-t connectivity
• edit distance
• longest common subsequence

Innovation

• catalytic algorithms
• sublinear space
• time-space tradeoff
• s-t connectivity
• edit distance