## 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

Funding Information:This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the ?Investments for the future? Programme IdEx Bordeaux ? CPU (ANR-10-IDEX-03-02), and also by the ANR-project ?DISPLEXITY?.

Publisher Copyright:

© 2016 Elsevier B.V.

## Keywords

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