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Deterministic Even-Cycle Detection in Broadcast CONGEST

Authors Pierre Fraigniaud , Maël Luce , Frédéric Magniez , Ioan Todinca

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  • Theory of computation → Distributed algorithms
Keywords
  • local computing
  • CONGEST model

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Abstract

We show that, for every k ≥ 2, C_{2k}-freeness can be decided in O(n^{1-(1)/(k)}) rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for k = 2 up to logarithmic factors thanks to the lower bound for C₄-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for k ∈ {3,4,5}, and by Fraigniaud et al. [PODC 2024] for k ≥ 6. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the "local density" of graphs without 2k-cycles, which we believe is interesting on its own.

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Pierre Fraigniaud, Maël Luce, Frédéric Magniez, and Ioan Todinca. Deterministic Even-Cycle Detection in Broadcast CONGEST. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 80:1-80:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.80

Author Details

Pierre Fraigniaud
  • Université Paris Cité, CNRS, IRIF, Paris, France
Maël Luce
  • Nagoya University, Japan
Frédéric Magniez
  • Université Paris Cité, CNRS, IRIF, Paris, France
Ioan Todinca
  • LIFO, Université d'Orléans and INSA Centre-Val de Loire, Orléans, France

Funding

Research supported in part by the European QuantERA project QOPT (ERA-NET Cofund 2022-25), the French ANR project DUCAT (ANR-20-CE48-0006), the French PEPR integrated project EPiQ (ANR-22-PETQ-0007), the French project FQPS (ANR‐21‐CMAQ-0001), and the Japanese grants JSPS KAKENHI 24H00071, JST ASPIRE JPMJAP2302 and MEXT Q-LEAP JPMXS0120319794.

References

  1. Stefan A Burr. Antidirected subtrees of directed graphs. Canadian Mathematical Bulletin, 25(1):119-120, 1982. Google Scholar
  2. Keren Censor-Hillel. Distributed subgraph finding: Progress and challenges. In 48th Int. Coll. on Automata, Languages, and Programming (ICALP), volume 198 of LIPIcs, pages 3:1-3:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  3. Keren Censor-Hillel, Orr Fischer, Tzlil Gonen, François Le Gall, Dean Leitersdorf, and Rotem Oshman. Fast distributed algorithms for girth, cycles and small subgraphs. In 34th International Symposium on Distributed Computing (DISC), volume 179 of LIPIcs, pages 33:1-33:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.DISC.2020.33.
  4. Andrew Drucker, Fabian Kuhn, and Rotem Oshman. On the power of the congested clique model. In 33rd ACM Symposium on Principles of Distributed Computing (PODC), pages 367-376, 2014. URL: https://doi.org/10.1145/2611462.2611493.
  5. Talya Eden, Nimrod Fiat, Orr Fischer, Fabian Kuhn, and Rotem Oshman. Sublinear-time distributed algorithms for detecting small cliques and even cycles. Distributed Computing, 35(3):207-234, June 2022. URL: https://doi.org/10.1007/S00446-021-00409-3.
  6. Pierre Fraigniaud, Maël Luce, Frédéric Magniez, and Ioan Todinca. Even-cycle detection in the randomized and quantum CONGEST model. In 43rd ACM Symposium on Principles of Distributed Computing (PODC), pages 209-219. ACM, 2024. URL: https://doi.org/10.1145/3662158.3662767.
  7. Pierre Fraigniaud, Maël Luce, Frédéric Magniez, and Ioan Todinca. Deterministic even-cycle detection in broadcast congest, 2025. Full version of this paper. URL: https://doi.org/10.48550/arXiv.2412.11195.
  8. Pierre Fraigniaud and Dennis Olivetti. Distributed detection of cycles. ACM Trans. Parallel Comput., 6(3):12:1-12:20, 2019. URL: https://doi.org/10.1145/3322811.
  9. Janne H. Korhonen and Joel Rybicki. Deterministic subgraph detection in broadcast CONGEST. In 21st International Conference on Principles of Distributed Systems (OPODIS), volume 95 of LIPIcs, pages 4:1-4:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.OPODIS.2017.4.
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