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Efficient Catalytic Graph Algorithms

Authors James Cook , Edward Pyne

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LIPIcs.ITCS.2026.43.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Complexity classes
Keywords
  • catalytic computing
  • graph algorithms
  • catalytic logspace

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ITCS.2026.43

Author Details

James Cook
  • Toronto, Canada
Edward Pyne
  • MIT, Cambridge, MA, USA

Funding

  • Pyne, Edward: Supported by NSF award CCF-2310818.

Acknowledgements

E.P. thanks Ryan Williams for encouragement to think about algorithms in CL and useful discussions, and Ian Mertz for the suggestion to work over different moduli. J.C. thanks Michal Koucký and the CCC reviewers for useful suggestions.

References

  1. Greg Barnes, Jonathan F. Buss, Walter L. Ruzzo, and Baruch Schieber. A sublinear space, polynomial time algorithm for directed s-t connectivity. SIAM J. Comput., 27(5):1273-1282, 1998. URL: https://doi.org/10.1137/S0097539793283151.
  2. Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff, and Florian Speelman. Computing with a full memory: catalytic space. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 857-866, New York, NY, USA, 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2591796.2591874.
  3. Kuan Cheng and William M. Hoza. Hitting sets give two-sided derandomization of small space. Adv. Math. Commun., 18:1-32, 2022. URL: https://doi.org/10.4086/TOC.2022.V018A021.
  4. James Cook. How to borrow memory. URL: https://www.falsifian.org/blog/2021/06/04/catalytic/.
  5. James Cook, Jiatu Li, Ian Mertz, and Edward Pyne. The structure of catalytic space: Capturing randomness and time via compression. Electronic Colloquium on Computational Complexity: ECCC, 2024. Google Scholar
  6. Dean Doron, Edward Pyne, and Roei Tell. Opening up the distinguisher: A hardness to randomness approach for BPL=L that uses properties of BPL. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 2039-2049, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649772.
  7. Michal Koucký, Ian Mertz, Edward Pyne, and Sasha Sami. Collapsing catalytic classes. Electron. Colloquium Comput. Complex., TR25-019, 2025. URL: https://eccc.weizmann.ac.il/report/2025/019.
  8. Ian Mertz. Reusing space: Techniques and open problems. Bulletin of EATCS, 141(3), 2023. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/780.
  9. Noam Nisan. On read once vs. multiple access to randomness in logspace. Theoretical Computer Science, 107(1):135-144, 1993. URL: https://doi.org/10.1016/0304-3975(93)90258-U.
  10. Edward Pyne. Derandomizing Logspace with a Small Shared Hard Drive. In Rahul Santhanam, editor, 39th Computational Complexity Conference (CCC 2024), volume 300 of Leibniz International Proceedings in Informatics (LIPIcs), pages 4:1-4:20, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.4.
  11. Walter J. Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences, 4:177-192, 1970. URL: https://doi.org/10.1016/S0022-0000(70)80006-X.
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