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Orientation Does Not Help with 3-Coloring a Grid in Online-LOCAL

Authors Thomas Boudier , Filippo Casagrande , Avinandan Das, Massimo Equi , Henrik Lievonen , Augusto Modanese , Ronja Stimpert

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LIPIcs.OPODIS.2025.19.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • coloring
  • locally checkable labeling problems
  • online algorithms

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Abstract

The online-LOCAL and SLOCAL models are extensions of the LOCAL model where nodes are processed in a sequential but potentially adversarial order. So far, the only problem we know of where the global memory of the online-LOCAL model has an advantage over SLOCAL is 3-coloring bipartite graphs. Recently, Chang et al. [PODC 2024] showed that even in grids, 3-coloring requires Ω(log n) locality in deterministic online-LOCAL. This result was subsequently extended by Akbari et al. [STOC 2025] to also hold in randomized online-LOCAL. However, both proofs heavily rely on the assumption that the algorithm does not have access to the orientation of the underlying grid. In this paper, we show how to lift this requirement and obtain the same lower bound (against either model) even when the algorithm is explicitly given a globally consistent orientation of the grid.

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Thomas Boudier, Filippo Casagrande, Avinandan Das, Massimo Equi, Henrik Lievonen, Augusto Modanese, and Ronja Stimpert. Orientation Does Not Help with 3-Coloring a Grid in Online-LOCAL. In 29th International Conference on Principles of Distributed Systems (OPODIS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 361, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.OPODIS.2025.19

Author Details

Thomas Boudier
  • Gran Sasso Science Institute, L’Aquila, Italy
Filippo Casagrande
  • Gran Sasso Science Institute, L’Aquila, Italy
Avinandan Das
  • Aalto University, Espoo, Finland
Massimo Equi
  • Aalto University, Espoo, Finland
Henrik Lievonen
  • Aalto University, Espoo, Finland
Augusto Modanese
  • Aalto University, Espoo, Finland
Ronja Stimpert
  • Aalto University, Espoo, Finland

Funding

This work was supported in part by the Research Council of Finland, Grants 359104 and 363558, as well as the Quantum Doctoral Education Pilot, the Ministry of Education and Culture, decision VN/3137/2024-OKM-4.

References

  1. Amirreza Akbari, Xavier Coiteux-Roy, Francesco D'Amore, François Le Gall, Henrik Lievonen, Darya Melnyk, Augusto Modanese, Shreyas Pai, Marc-Olivier Renou, Václav Rozhon, and Jukka Suomela. Online Locality Meets Distributed Quantum Computing. In Michal Koucký and Nikhil Bansal, editors, Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, Prague, Czechia, June 23-27, 2025, pages 1295-1306. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718211.
  2. Amirreza Akbari, Navid Eslami, Henrik Lievonen, Darya Melnyk, Joona Särkijärvi, and Jukka Suomela. Locality in Online, Dynamic, Sequential, and Distributed Graph Algorithms. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 10:1-10:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.10.
  3. Jack E. Bresenham. Algorithm for computer control of a digital plotter. IBM Systems Journal, 4(1):25-30, 1965. URL: https://doi.org/10.1147/sj.41.0025.
  4. Yi-Jun Chang, Gopinath Mishra, Hung Thuan Nguyen, Mingyang Yang, and Yu-Cheng Yeh. A Tight Lower Bound for 3-Coloring Grids in the Online-LOCAL Model. In Ran Gelles, Dennis Olivetti, and Petr Kuznetsov, editors, Proceedings of the 43rd ACM Symposium on Principles of Distributed Computing, PODC 2024, Nantes, France, June 17-21, 2024, pages 106-116. ACM, 2024. URL: https://doi.org/10.1145/3662158.3662794.
  5. Mohsen Ghaffari, Fabian Kuhn, and Yannic Maus. On the complexity of local distributed graph problems. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 784-797. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055471.
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