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DOI: 10.4230/LIPIcs.ITCS.2026.43

Efficient Catalytic Graph Algorithms

Authors: James Cook and Edward Pyne

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We give fast, simple, and implementable catalytic logspace algorithms for two fundamental graph problems. First, a randomized catalytic algorithm for s → t connectivity running in Õ(nm) time, and a deterministic catalytic algorithm for the same running in Õ(n³ m) time. The former algorithm is the first algorithmic use of randomization in CL. The algorithm uses one register per vertex and repeatedly "pushes" values along the edges in the graph. Second, a deterministic catalytic algorithm for simulating random walks which in Õ(m T² / ε) time estimates the probability a T-step random walk ends at a given vertex within ε additive error. The algorithm uses one register for each vertex and increments it at each visit to ensure repeated visits follow different outgoing edges. Prior catalytic algorithms for both problems did not have explicit runtime bounds beyond being polynomial in n.

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James Cook and Edward Pyne. Efficient Catalytic Graph Algorithms. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 43:1-43:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cook_et_al:LIPIcs.ITCS.2026.43,
  author =	{Cook, James and Pyne, Edward},
  title =	{{Efficient Catalytic Graph Algorithms}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{43:1--43:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.43},
  URN =		{urn:nbn:de:0030-drops-253305},
  doi =		{10.4230/LIPIcs.ITCS.2026.43},
  annote =	{Keywords: catalytic computing, graph algorithms, catalytic logspace}
}

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DOI: 10.4230/LIPIcs.CCC.2022.8

Trading Time and Space in Catalytic Branching Programs

Authors: James Cook and Ian Mertz

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

An m-catalytic branching program (Girard, Koucký, McKenzie 2015) is a set of m distinct branching programs for f which are permitted to share internal (i.e. non-source non-sink) nodes. While originally introduced as a non-uniform analogue to catalytic space, this also gives a natural notion of amortized non-uniform space complexity for f, namely the smallest value |G|/m for an m-catalytic branching program G for f (Potechin 2017). Potechin (2017) showed that every function f has amortized size O(n), witnessed by an m-catalytic branching program where m = 2^(2ⁿ-1). We recreate this result by defining a catalytic algorithm for evaluating polynomials using a large amount of space but O(n) time. This allows us to balance this with previously known algorithms which are efficient with respect to space at the cost of time (Cook, Mertz 2020, 2021). We show that for any ε ≥ 2n^(-1), every function f has an m-catalytic branching program of size O_ε(mn), where m = 2^(2^(ε n)). We similarly recreate an improved result due to Robere and Zuiddam (2021), and show that for d ≤ n and ε ≥ 2d^(-1), the same result holds for m = 2^binom(n, ≤ ε d) as long as f is a degree-d polynomial over 𝔽₂. We also show that for certain classes of functions, m can be reduced to 2^(poly n) while still maintaining linear or quasi-linear amortized size. In the other direction, we bound the necessary length, and by extension the amortized size, of any permutation branching program for an arbitrary function between 3n and 4n-4.

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James Cook and Ian Mertz. Trading Time and Space in Catalytic Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cook_et_al:LIPIcs.CCC.2022.8,
  author =	{Cook, James and Mertz, Ian},
  title =	{{Trading Time and Space in Catalytic Branching Programs}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{8:1--8:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.8},
  URN =		{urn:nbn:de:0030-drops-165708},
  doi =		{10.4230/LIPIcs.CCC.2022.8},
  annote =	{Keywords: complexity theory, branching programs, amortized, space complexity, catalytic computation}
}

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Documents authored by Cook, James

Efficient Catalytic Graph Algorithms

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Trading Time and Space in Catalytic Branching Programs

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