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

Roman Edenhofer

TL;DR

A space-space trade-off for directed $st$-connectivity in the catalytic space model is proved and the techniques extended to counting the number of walks from $s$ to $t$ of a given length $\ell\leq n$.

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 $\ell\leq n$.

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

TL;DR

A space-space trade-off for directed -connectivity in the catalytic space model is proved and the techniques extended to counting the number of walks from to of a given length .

Abstract

We prove a space-space trade-off for directed -connectivity in the catalytic space model. For any integer , we give an algorithm that decides directed -connectivity using regular workspace and bits of catalytic memory. This interpolates between the classical -space bound from Savitch's algorithm and a catalytic endpoint with workspace and 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 to of a given length .
Paper Structure (6 sections, 6 theorems, 33 equations, 6 algorithms)

This paper contains 6 sections, 6 theorems, 33 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Catalytic Space and Register Programs
  3. Our Techniques
  4. A Catalytic Variant of Savitch's Algorithm
  5. Our Space-space Trade-off
  6. Dealing with Invalid Initial Registers

Key Result

theorem 1

Given a directed graph $G$ of size $n$, vertices $s,t\in V(G)$, a length $\ell\in[n]$ and a parameter $k\in[n]$, there is an algorithm running in $\mathsf{CSPACE}(O(\log \ell\cdot\log k+\log n),\;O(\frac{n}{k}\cdot\log^2 n))$ that outputs the number of length-$\ell$ walks from $s$ to $t$. In particu

Theorems & Definitions (13)

  • theorem 1
  • corollary 1
  • proof
  • proof
  • proof
  • lemma 1: CDL01
  • theorem 2
  • proof
  • proof
  • proof
  • ...and 3 more