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 l-colorable is called a strong resp., weak d, l-network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. Awerbuch et al. showed that strong $exp\{O\sqrt{ \log n \log \log n}\}$, $exp\{O\sqrt{ \log n \log \log n}\}$-network-decompositions can be computed in distributed deterministic time $exp\{O\sqrt{ \log n \log \log n}\}$. 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=O1. However, the parameter l in his result is On1/2+e. In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong O1, One-network-decompositions. As a corollary we derive a constant-time randomized One-approximation algorithm for the distributed minimum coloring problem. This improves the best previously-known On1/2+e 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 polylogarithmic -time algorithm. We devise a deterministic polylogarithmic-time approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer 2010.