Making Self-Stabilizing Algorithms for Any Locally Greedy Problem
Self-stabilizing algorithms are a way to deal with network dynamicity, as it will update itself after a network change (addition or removal of nodes or edges), as long as changes are not frequent. We propose an automatic transformation of synchronous distributed algorithms that solve locally greedy and mendable problems into self-stabilizing algorithms in anonymous networks.
Mendable problems are a generalization of greedy problems where any partial solution may be transformed -instead of completed- into a global solution: every time we extend the partial solution, we are allowed to change the previous partial solution up to a given distance. Locally here means that to extend a solution for a node, we need to look at a constant distance from it.
In order to do this, we propose the first explicit self-stabilizing algorithm computing a (k,k-1)-ruling set (i.e. a "maximal independent set at distance k"). By combining this technique multiple times, we compute a distance-K coloring of the graph. With this coloring we can finally simulate Local model algorithms running in a constant number of rounds, using the colors as unique identifiers.
Our algorithms work under the Gouda daemon, similar to the probabilistic daemon: if an event should eventually happen, it will occur.
Greedy Problem
Ruling Set
Distance-K Coloring
Self-Stabilizing Algorithm
Theory of computation~Distributed algorithms
11:1-11:17
Regular Paper
https://arxiv.org/abs/2208.14700
Johanne
Cohen
Johanne Cohen
Université Paris-Saclay, CNRS, LISN, 91405, Orsay, France
https://orcid.org/0000-0002-9548-5260
Laurence
Pilard
Laurence Pilard
LI-PaRAD, UVSQ, Université Paris-Saclay, France
https://orcid.org/0000-0002-1104-8216
Mikaël
Rabie
Mikaël Rabie
IRIF-CNRS, Université Paris Cité, France
Jonas
Sénizergues
Jonas Sénizergues
Université Paris-Saclay, CNRS, LISN, 91405, Orsay, France
10.4230/LIPIcs.SAND.2023.11
Yehuda Afek and Shlomi Dolev. Local stabilizer. Journal of Parallel and Distributed Computing, 62(5):745-765, 2002.
Yehuda Afek, Shay Kutten, and Moti Yung. Local detection for global self stabilization. Theoretical Computer Science, 186(1-2):339, 1991.
Karine Altisen, Stéphane Devismes, Swan Dubois, and Franck Petit. Introduction to distributed self-stabilizing algorithms. Synthesis Lectures on Distributed Computing Theory, 8(1):1-165, 2019.
Baruch Awerbuch. Complexity of network synchronization. J. ACM, 32(4):804-823, 1985.
Baruch Awerbuch, Andrew V Goldberg, Michael Luby, and Serge A Plotkin. Network decomposition and locality in distributed computation. In FOCS, volume 30, pages 364-369. Citeseer, 1989.
Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. Lower bounds for maximal matchings and maximal independent sets. Journal of the ACM (JACM), 68(5):1-30, 2021.
Alkida Balliu, Sebastian Brandt, and Dennis Olivetti. Distributed lower bounds for ruling sets. SIAM Journal on Computing, 51(1):70-115, 2022.
Alkida Balliu, Juho Hirvonen, Darya Melnyk, Dennis Olivetti, Joel Rybicki, and Jukka Suomela. Local mending. In International Colloquium on Structural Information and Communication Complexity, pages 1-20. Springer, 2022.
Leonid Barenboim, Michael Elkin, and Uri Goldenberg. Locally-iterative distributed (δ+ 1) -coloring below szegedy-vishwanathan barrier, and applications to self-stabilization and to restricted-bandwidth models. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, pages 437-446, 2018.
Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed symmetry breaking. Journal of the ACM (JACM), 63(3):1-45, 2016.
Shimon Bitton, Yuval Emek, Taisuke Izumi, and Shay Kutten. Fully adaptive self-stabilizing transformer for lcl problems. arXiv preprint, 2021. URL: https://arxiv.org/abs/2105.09756.
https://arxiv.org/abs/2105.09756
Sebastian Brandt, Juho Hirvonen, Janne H Korhonen, Tuomo Lempiäinen, Patric RJ Östergård, Christopher Purcell, Joel Rybicki, Jukka Suomela, and Przemysław Uznański. Lcl problems on grids. In Proceedings of the ACM Symposium on Principles of Distributed Computing, pages 101-110, 2017.
Keren Censor-Hillel, Merav Parter, and Gregory Schwartzman. Derandomizing local distributed algorithms under bandwidth restrictions. Distributed Computing, 33(3):349-366, 2020.
Richard Cole and Uzi Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control, 70(1):32-53, 1986.
Alain Cournier, AK Datta, Franck Petit, and Vincent Villain. Self-stabilizing pif algorithm in arbitrary rooted networks. In Proceedings 21st International Conference on Distributed Computing Systems, pages 91-98. IEEE, 2001.
Alain Cournier, Stéphane Devismes, and Vincent Villain. Snap-stabilizing pif and useless computations. In 12th International Conference on Parallel and Distributed Systems-(ICPADS'06), volume 1, pages 8-pp. IEEE, 2006.
Shlomi Dolev. Self-stabilization. MIT press, 2000.
Swan Dubois and Sébastien Tixeuil. A taxonomy of daemons in self-stabilization. arXiv preprint, 2011. URL: https://arxiv.org/abs/1110.0334.
https://arxiv.org/abs/1110.0334
Mohsen Ghaffari. An improved distributed algorithm for maximal independent set. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, pages 270-277. SIAM, 2016.
Mohsen Ghaffari, Christoph Grunau, and Václav Rozhoň. Improved deterministic network decomposition. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2904-2923. SIAM, 2021.
Mohsen Ghaffari, Fabian Kuhn, and Yannic Maus. On the complexity of local distributed graph problems. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 784-797, 2017.
Sukumar Ghosh, Arobinda Gupta, Ted Herman, and Sriram V Pemmaraju. Fault-containing self-stabilizing algorithms. In Proceedings of the fifteenth annual ACM symposium on Principles of distributed computing, pages 45-54, 1996.
Wayne Goddard, Stephen T Hedetniemi, David Pokrass Jacobs, and Pradip K Srimani. Self-stabilizing protocols for maximal matching and maximal independent sets for ad hoc networks. In Proceedings International Parallel and Distributed Processing Symposium, pages 14-pp. IEEE, 2003.
Mohamed G Gouda. The theory of weak stabilization. In International Workshop on Self-Stabilizing Systems, pages 114-123. Springer, 2001.
Nabil Guellati and Hamamache Kheddouci. A survey on self-stabilizing algorithms for independence, domination, coloring, and matching in graphs. Journal of Parallel and Distributed Computing, 70(4):406-415, 2010.
Stephen T Hedetniemi. Self-stabilizing domination algorithms. Structures of Domination in Graphs, pages 485-520, 2021.
Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. SIAM Journal on Computing, 50(3):STOC16-98, 2019.
Michiyo Ikeda, Sayaka Kamei, and Hirotsugu Kakugawa. A space-optimal self-stabilizing algorithm for the maximal independent set problem. In the Third International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT), pages 70-74. Citeseer, 2002.
Shay Kutten and David Peleg. Fault-local distributed mending. Journal of Algorithms, 30(1):144-165, 1999.
Shay Kutten and David Peleg. Tight fault locality. SIAM Journal on Computing, 30(1):247-268, 2000.
Moni Naor and Larry Stockmeyer. What can be computed locally? SIAM Journal on Computing, 24(6):1259-1277, 1995.
Alessandro Panconesi and Aravind Srinivasan. The local nature of δ-coloring and its algorithmic applications. Combinatorica, 15(2):255-280, 1995.
David Peleg. Distributed computing: a locality-sensitive approach. SIAM, 2000.
Václav Rozhoň and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 350-363, 2020.
Sandeep K Shukla, Daniel J Rosenkrantz, S Sekharipuram Ravi, et al. Observations on self-stabilizing graph algorithms for anonymous networks. In Proceedings of the second workshop on self-stabilizing systems, volume 7, page 15, 1995.
Jukka Suomela. Survey of local algorithms. ACM Computing Surveys (CSUR), 45(2):1-40, 2013.
Volker Turau. Linear self-stabilizing algorithms for the independent and dominating set problems using an unfair distributed scheduler. Information Processing Letters, 103(3):88-93, 2007.
Volker Turau. Computing fault-containment times of self-stabilizing algorithms using lumped markov chains. Algorithms, 11(5):58, 2018.
Volker Turau. Making randomized algorithms self-stabilizing. In International Colloquium on Structural Information and Communication Complexity, pages 309-324. Springer, 2019.
Volker Turau and Christoph Weyer. Randomized self-stabilizing algorithms for wireless sensor networks. In Self-Organizing Systems, pages 74-89. Springer, 2006.
Johanne Cohen, Laurence Pilard, Mikaël Rabie, and Jonas Sénizergues
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode