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Distributed Recoloring of Interval and Chordal Graphs

Authors Nicolas Bousquet , Laurent Feuilloley , Marc Heinrich , Mikaël Rabie

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Author Details

Nicolas Bousquet
  • Univ. Lyon, Université Lyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France
Laurent Feuilloley
  • Univ. Lyon, Université Lyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France
Marc Heinrich
  • University of Leeds, UK
Mikaël Rabie
  • Université de Paris, CNRS, IRIF, F-75013, Paris, France


We thank the reviewers for their useful comments, and Marthe Bonamy for starting this project with us. The second author thanks Fabian Kuhn and Václav Rozhoň for their kind and expert answers to his questions on network decomposition.

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Nicolas Bousquet, Laurent Feuilloley, Marc Heinrich, and Mikaël Rabie. Distributed Recoloring of Interval and Chordal Graphs. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 19:1-19:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


One of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions. First, the problem of recoloring, that is computing an efficient transformation between two given colorings (instead of computing a new coloring), has been considered, both to model radio network updates, and as a useful subroutine for coloring. Second, as it appears that general graphs and bounded-degree graphs do not model real networks very well (with, respectively, pathological worst-case topologies and too strong assumptions), coloring has been studied in more specific graph classes. In this paper, we study the intersection of these two directions: distributed recoloring in two relevant graph classes, interval and chordal graphs. More formally, the question of recoloring a graph is as follows: we are given a network, an input coloring α and a target coloring β, and we want to find a schedule of colorings to reach β starting from α. In a distributed setting, the schedule needs to be found within the LOCAL model, where nodes communicate with their direct neighbors synchronously. The question we want to answer is: how many rounds of communication {are} needed to produce a schedule, and what is the length of this schedule? In the case of interval and chordal graphs, we prove that, if we have less than 2ω colors, ω being the size of the largest clique, extra colors will be needed in the intermediate colorings. For interval graphs, we produce a schedule after O(poly(Δ)log*n) rounds of communication, and for chordal graphs, we need O(ω²Δ²log n) rounds to get one. Our techniques also improve classic coloring algorithms. Namely, we get ω+1-colorings of interval graphs in O(ωlog*n) rounds and of chordal graphs in O(ωlog n) rounds, which improves on previous known algorithms that use ω+2 colors for the same running times.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Distributed coloring
  • distributed recoloring
  • interval graphs
  • chordal graphs
  • intersection graphs


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  1. Pierre Aboulker, Marthe Bonamy, Nicolas Bousquet, and Louis Esperet. Distributed coloring in sparse graphs with fewer colors. Electron. J. Comb., 26(4):P4.20, 2019. Google Scholar
  2. Leonid Barenboim. On the locality of some np-complete problems. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, ICALP 2012, volume 7392, pages 403-415, 2012. URL:
  3. Leonid Barenboim and Michael Elkin. Sublogarithmic distributed MIS algorithm for sparse graphs using nash-williams decomposition. Distributed Comput., 22(5-6):363-379, 2010. URL:
  4. Leonid Barenboim and Michael Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers, 2013. URL:
  5. Leonid Barenboim, Michael Elkin, and Cyril Gavoille. A fast network-decomposition algorithm and its applications to constant-time distributed computation. Theor. Comput. Sci., 751:2-23, 2018. URL:
  6. Jean RS Blair and Barry Peyton. An introduction to chordal graphs and clique trees. In Graph theory and sparse matrix computation, pages 1-29. Springer, 1993. Google Scholar
  7. Marthe Bonamy and Nicolas Bousquet. Recoloring graphs via tree decompositions. Eur. J. Comb., 69:200-213, 2018. URL:
  8. Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, and Daniël Paulusma. Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. J. Comb. Optim., 27(1):132-143, 2014. URL:
  9. Marthe Bonamy, Paul Ouvrard, Mikaël Rabie, Jukka Suomela, and Jara Uitto. Distributed recoloring. In DISC 2018, volume 121, pages 12-1, 2018. Google Scholar
  10. Nicolas Bousquet and Valentin Bartier. Linear transformations between colorings in chordal graphs. In ESA 2019, volume 144 of LIPIcs, pages 24:1-24:15, 2019. URL:
  11. Nicolas Bousquet, Laurent Feuilloley, Marc Heinrich, and Mikaël Rabie. Distributed recoloring of interval and chordal graphs. CoRR, abs/2109.06021, 2021. URL:
  12. Nicolas Bousquet and Marc Heinrich. A polynomial version of cereceda’s conjecture. CoRR, abs/1903.05619, 2019. Google Scholar
  13. Keren Censor-Hillel and Mikaël Rabie. Distributed reconfiguration of maximal independent sets. Journal of Computer and System Sciences, 112:85-96, 2020. Google Scholar
  14. L. Cereceda. Mixing Graph Colourings. PhD thesis, London School of Economics and Political Science, 2007. Google Scholar
  15. Shiri Chechik and Doron Mukhtar. Optimal distributed coloring algorithms for planar graphs in the LOCAL model. In SODA 2019, pages 787-804. SIAM, 2019. URL:
  16. Richard Cole and Uzi Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Inf. Control., 70(1):32-53, 1986. URL:
  17. M. Dyer, A. D. Flaxman, A. M Frieze, and E. Vigoda. Randomly coloring sparse random graphs with fewer colors than the maximum degree. Random Structures & Algorithms, 29(4):450-465, 2006. Google Scholar
  18. Mohsen Ghaffari. Network decomposition and distributed derandomization (invited paper). In SIROCCO 2020, volume 12156, pages 3-18, 2020. URL:
  19. Mohsen Ghaffari and Christiana Lymouri. Simple and near-optimal distributed coloring for sparse graphs. In DISC 2017, volume 91 of LIPIcs, pages 20:1-20:14, 2017. URL:
  20. Andrew V. Goldberg, Serge A. Plotkin, and Gregory E. Shannon. Parallel symmetry-breaking in sparse graphs. SIAM J. Discret. Math., 1(4):434-446, 1988. URL:
  21. Martin Charles Golumbic. Algorithmic graph theory and perfect graphs. Elsevier, 1980. Google Scholar
  22. Magnús M. Halldórsson and Christian Konrad. Distributed algorithms for coloring interval graphs. In DISC 2014, volume 8784, pages 454-468, 2014. URL:
  23. Magnús M. Halldórsson and Christian Konrad. Improved distributed algorithms for coloring interval graphs with application to multicoloring trees. Theor. Comput. Sci., 811:29-41, 2020. URL:
  24. Christian Konrad and Viktor Zamaraev. Distributed minimum vertex coloring and maximum independent set in chordal graphs. In MFCS 2019, volume 138 of LIPIcs, pages 21:1-21:15, 2019. URL:
  25. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. On the locality of bounded growth. In PODC 2005, pages 60-68. ACM, 2005. URL:
  26. Nathan Linial. Locality in distributed graph algorithms. SIAM J. Comput., 21(1):193-201, 1992. URL:
  27. Frédéric Maffray. On the coloration of perfect graphs. In Recent Advances in Algorithms and Combinatorics, pages 65-84. Springer, 2003. Google Scholar
  28. Gary L. Miller and John H. Reif. Parallel tree contraction part 1: Fundamentals. Adv. Comput. Res., 5:47-72, 1989. Google Scholar
  29. Alessandro Panconesi and Aravind Srinivasan. The local nature of δ-coloring and its algorithmic applications. Combinatorica, 15(2):255-280, 1995. Google Scholar
  30. Marinus Johannes Petrus Peeters. On coloring j-unit sphere graphs. Technical report, Tilburg University, School of Economics and Management, 1991. Google Scholar
  31. Ram Ramanathan. A unified framework and algorithm for channel assignment in wireless networks. Wirel. Networks, 5(2):81-94, 1999. URL:
  32. Johannes Schneider and Roger Wattenhofer. An optimal maximal independent set algorithm for bounded-independence graphs. Distributed Comput., 22(5-6):349-361, 2010. URL:
  33. Lieven Vandenberghe and Martin S. Andersen. Chordal graphs and semidefinite optimization. Found. Trends Optim., 1(4):241-433, 2015. URL:
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