A Space-space Trade-off for Directed st-Connectivity

TL;DR

This paper introduces a new space-space trade-off algorithm for directed st-connectivity in the catalytic space model, balancing regular workspace and catalytic memory to interpolate between classical and catalytic bounds.

Contribution

It provides the first algorithm achieving a tunable trade-off between regular workspace and catalytic memory for directed st-connectivity, extending to walk counting.

Findings

Achieves a space-catalytic memory trade-off for directed st-connectivity.

Matches the smallest known catalytic logspace size up to logarithmic factors.

Extends techniques to counting walks of bounded length.

Abstract

We prove a space-space trade-off for directed $st$-connectivity in the catalytic space model. For any integer $k \leq n$, we give an algorithm that decides directed $st$-connectivity using $O(\log n \cdot \log k+\log n)$ regular workspace and $O\left(\frac{n}{k} \cdot \log^2 n\right)$ bits of catalytic memory. This interpolates between the classical $O(\log^2 n)$-space bound from Savitch's algorithm and a catalytic endpoint with $O(\log n)$ workspace and $O(n\cdot \log^2 n)$ catalytic memory. As a warm-up, we present a catalytic variant of Savitch's algorithm achieving the endpoint above. Up to logarithmic factors, this matches the smallest catalyst size currently known for catalytic logspace algorithms, due to Cook and Pyne (ITCS 2026). Our techniques also extend to counting the number of walks from $s$ to $t$ of a given length $\ell\leq n$.