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Frontier Space-Time Algorithms Using Only Full Memory

Petr Chmel, Aditi Dudeja, Michal Koucký, Ian Mertz, Ninad Rajgopal

TL;DR

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, and improves the catalytic space usage of the best known algorithm.

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 $n \big/ 2^{Θ(\sqrt{\log n})}$ 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 Fréchet Distance, again using $n \big/ 2^{Θ(\sqrt{\log n})}$ catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].

Frontier Space-Time Algorithms Using Only Full Memory

TL;DR

A polynomial time algorithm for directed -n \big/ 2^{\Theta(\sqrt{\log n})}$ catalytic space, which matches the state-of-the-art time-space bounds in the non-catalytic setting, and improves the catalytic space usage of the best known algorithm.

Abstract

We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only 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 - connectivity using 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 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 Fréchet Distance, again using catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].
Paper Structure (25 sections, 21 theorems, 35 equations, 4 figures)

This paper contains 25 sections, 21 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Catalytic Computation
  3. Sublinear-Space Polynomial-Time Algorithms
  4. Our Results
  5. Proof Overview
  6. Previous algorithms for $\mathsf{STCONN}$: BBRS98 & cook_pyne_26_efficient
  7. Long and Short Paths Framework (BBRS98).
  8. Our approach for $\mathsf{STCONN}$
  9. $\mathsf{ED}$, $\mathsf{LCS}$, & $\mathsf{DFD}$ Without Direct Reductions To $\mathsf{STCONN}$
  10. Future Work
  11. Preliminaries
  12. Catalytic Model
  13. Function Statements
  14. Pairwise Independent Hash Functions
  15. Modular Arithmetic
  16. ...and 10 more sections

Key Result

Theorem 1.1

There exists a deterministic algorithm solving $\mathsf{STCONN}$ on $n$-vertex graphs in time $\mathsf{poly}(n)$, space $\mathcal{O}(\log(n))$, and catalytic space $\frac{n}{2^{\Omega{\left( \sqrt{\log(n)} \right)}}}$. Furthermore, for every $\epsilon > 0$, there is a randomized algorithm that runs

Figures (4)

  • Figure 1: An example of $\mathsf{Grid}_{5,5}$ with three color classes.
  • Figure 2: The larger red vertices are a part of the induced rectangle, but the bold red path connecting them leaves the induced rectangle.
  • Figure 3: The behavior of the masking oracle in \ref{['lem:oracle_transform_subgraph']}. The dashed edges have their edge weights set to $0$ by the mask.
  • Figure 4: The transformation from $\mathsf{ED}_{n,n}$ into $\mathsf{Grid}_{2n,2n}$. The source and sink vertices of $\mathsf{ED}_{n,n}$ and $\mathsf{Grid}_{2n,2n}$ which correspond to each other are shown in the same color. There are no edges in $\mathsf{Grid}_{2n,2n}$ outside the dashed square.

Theorems & Definitions (45)

  • Theorem 1.1: Sublinear catalytic space algorithms for $\mathsf{STCONN}$
  • Theorem 1.2: Sub-linear catalytic space algorithms for $\mathsf{ED}, \mathsf{LCS}$, and $\mathsf{DFD}$
  • Definition 2.1: Catalytic Turing machine
  • Definition 2.2: Catalytic subroutine
  • Definition 2.3: Connectivity
  • Definition 2.4: Longest common subsequence
  • Definition 2.5: Edit distance
  • Definition 2.6: Discrete Fréchet distance
  • Definition 2.7: Pairwise Independent Hash Functions
  • Lemma 2.8: vadhan_psuedo_2012
  • ...and 35 more