A Fast Network-Decomposition Algorithm and its Applications to Constant-Time Distributed Computation

A partition (C1, C2, ..., Cq) ofG = (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(√logn log logn)}, exp{O(√logn log logn)})-network-decompositions can be computed in distributed deterministic time exp{O(√logn log logn)}. 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. The result of Awerbuch et al. was improved by Panconesi and Srinivasan in 1992: in the latter result d = l = exp{O(√logn)}, and the running time is exp{O(√logn)} as well. In another remarkable breakthrough Linial and Saks (in 1992) showed that weak (O(log n), O(log n))-network-decompositions can be computed in distributed randomized time O(log n). Much more recently Barenboim (2012) devised a distributed randomized constant-time algorithm for computing strong network decompositions with d = O(1). However, the parameter l 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, improving the previously best-known 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 polylogarithmic-time algorithm. We devise a deterministic polylogarithmic-time approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer (2010).