## Abstract

For a pair of positive parameters D,x, a partition P of the vertex set V of an n-vertex graph G = (V,E) into disjoint clusters of diameter at most D each is called a (D,x) net- work decomposition, if the supergraph G(P), obtained by contracting each of the clusters of P, can be properly x- colored. The decomposition P is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diame- ter at most D, i.e., if for every cluster C ϵ P and every two vertices u; v ϵ C, the distance between them in the induced graph G(C) of C (resp., in G) is at most D. Network decomposition is a powerful construct, very use- ful in distributed computing and beyond. It was introduced by Awerbuch et. al. [AGLP89] in the end of the eighties. These authors showed that strong (2^{O}(√log n log log n); 2^{O}(√log n log log n)) network decompositions can be computed in 2^{O}(√log n log log n) distributed time. Their result was improved at the beginning of nineties by Pan- conesi and Srinivasan [PS92], who showed that 2^{O}(√log n log log n) in all the three expressions can be replaced by 2^{O}(√log n). Around the same time Linial and Saks [LS93] devised an ingenious randomized algorithm that constructs weak (O(log n);O(log n)) network decompositions in O(log^{2} n) time. Awerbuch et. al. [ABCP96] devised a randomized al- gorithm that builds a strong (O(log n);O(log n)) network decomposition in O(log^{4} n) time, using very large messages and heavy local computations. It was however open till now if strong network decompositions with both parame-ters 2^{O}(√log n) can be constructed in distributed 2^{O}(√log n) time using short messages, or if a result of [LS93] can be strengthened to provide a strong (O(log n);O(log n)) net- work decomposition within O(log^{2} n) time (even using large messages). In this paper we answer these long-standing open ques- tions in the afirmative, and show that strong (O(log n);O(log n)) network decompositions can be computed in O(log^{2} n) time. We also present a tradeo between pa- rameters of our network decomposition. Our work is inspired by and relies on the \shifted shortest path approach", due to Blelloch et. al. [BGK+14], and Miller et. al. [MPX13]. These authors developed this approach for PRAM algorithms for padded partitions. We adapt their approach to network decompositions in the distributed model of computation.

Original language | English |
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Title of host publication | PODC 2016 - Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing |

Publisher | Association for Computing Machinery |

Pages | 211-216 |

Number of pages | 6 |

ISBN (Electronic) | 9781450339643 |

DOIs | |

State | Published - 25 Jul 2016 |

Event | 35th ACM Symposium on Principles of Distributed Computing, PODC 2016 - Chicago, United States Duration: 25 Jul 2016 → 28 Jul 2016 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Principles of Distributed Computing |
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Volume | 25-28-July-2016 |

### Conference

Conference | 35th ACM Symposium on Principles of Distributed Computing, PODC 2016 |
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Country/Territory | United States |

City | Chicago |

Period | 25/07/16 → 28/07/16 |

## Keywords

- Distributed Model
- Network decompositions

## ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications