## Abstract

An (α,β)-spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u,v, d _{H} (u,v) ≤α•d _{G} (u,v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α,β)-spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every n-node graph and integer kge;1, an (α,β)- spanner of O(βn ^{1+1/k} ) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k-1,0)-spanner of at most k n ^{1+1/k} edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC '08). For k=2 and constant ε>0, it can also produce a (1+ε,2-ε)-spanner of O(n ^{3/2}) edges in constant time. More interestingly, for every integer k>1, it can construct in constant time a (1+ε,O(1/ε) ^{k-2})-spanner of O(ε ^{-k+1} n ^{1+1/k} ) edges. Such deterministic construction was not previously known. The paper also presents a second generic deterministic and distributed algorithm based on the construction of small dominating sets and maximal independent sets. After computing such sets in sub-polynomial time, it constructs at its best a (1+ε,β)-spanner with O(βn ^{1+1/k} ) edges, where β=k ^{log(logk/ε)+O(1)}. For k=3, it provides a (1+ε, 6-ε)-spanner with O(ε ^{-1} n ^{4/3}) edges. The additive terms β=β(k,ε) in the stretch of our constructions yield the best trade-off currently known between k and ε, due to Elkin and Peleg (STOC '01). Our distributed algorithms are rather short, and can be viewed as a unification and simplification of previous constructions.

Original language | English |
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Title of host publication | Distributed Computing - 23rd International Symposium, DISC 2009, Proceedings |

Publisher | Springer Verlag |

Pages | 176-190 |

Number of pages | 15 |

ISBN (Print) | 3642043542, 9783642043543 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

Event | 23rd International Symposium on Distributed Computing, DISC 2009 - Elche, Spain Duration: 23 Sep 2009 → 25 Sep 2009 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5805 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 23rd International Symposium on Distributed Computing, DISC 2009 |
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Country/Territory | Spain |

City | Elche |

Period | 23/09/09 → 25/09/09 |