A Fast Network-Decomposition Algorithm and Its Applications to Constant-Time Distributed Computation - (Extended Abstract)

A partition (C1, C2, ..., Cq) of G = (V,E) into clusters of strong (respectively, weak) diameter d, such that the supergraph obtained by contracting each Ci is -colorable is called a strong (resp., weak) (d, )-network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. Awerbuch et al. showed that strong (exp{O(√log n log log n)}, exp{O(√log n log log n)})-network-decompositions can be computed in distributed deterministic time exp{O(√log n log log n)}. Even more importantly, they demonstrated that network-decompositions can be used for a great variety of applications in the message-passing model of distributed computing. Much more recently Barenboim (2012) devised a distributed randomized constant-time algorithm for computing strong network decompositions with d = O(1). However, the parameter in his result is O(n ). In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong (O(1), O(n ))-network-decompositions. As a corollary we derive a constant-time randomized O(n )-approximation algorithm for the distributed minimum coloring problem. This improves the best previouslyknown O(n ) approximation guarantee. We also derive other improved distributed algorithms for a variety of problems. Most notably, for the extremely well-studied distributed minimum dominating set problem currently there is no known deterministic polyA full version of this paper with all proofs omitted from the current version due to lack of space is available online [10]. Part of this work has been performed while the author was a postdoctoral fellow at a joint program of the Simons Institute at UC Berkeley and I-Core at Weizmann Institute. This research has been supported by the Israeli Academy of Science, grant 593/11, and by the Binational Science Foundation, grant 2008390. c © Springer International Publishing Switzerland 2015 C. Scheideler (Ed.): SIROCCO 2015, LNCS 9439, pp. 209–223, 2015. DOI: 10.1007/978-3-319-25258-2_15 210 L. Barenboim, M. Elkin, and C. Gavoille logarithmic -time algorithm. We devise a deterministic polylogarithmictime approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer (2010).