## Abstract

A partition (C_{1},C_{2},…,C_{q}) of G=(V,E) into clusters of strong (respectively, weak) diameter d, such that the supergraph obtained by contracting each C_{i} 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 (d,ℓ)-network-decompositions with d=ℓ=exp{O(lognloglogn)} can be computed in distributed deterministic time O(d). 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 O(d) as well. In another remarkable breakthrough Linial and Saks (in 1992) showed that weak (O(logn),O(logn))-network-decompositions can be computed in distributed randomized time O(log^{2}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^{1/2+ϵ}). 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^{ϵ}))-network-decompositions. As a corollary we derive a constant-time randomized O(n^{ϵ})-approximation algorithm for the distributed minimum coloring problem, improving the previously best-known O(n^{1/2+ϵ}) 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).

Original language | English |
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Pages (from-to) | 2-23 |

Number of pages | 22 |

Journal | Theoretical Computer Science |

Volume | 751 |

DOIs | |

State | Published - 3 Dec 2018 |

### Bibliographical note

Publisher Copyright:© 2016 Elsevier B.V.

## Keywords

- Coloring
- Distributed algorithms
- Dominating sets
- Local algorithms
- Spanners