## Abstract

Graph spanners are sparse subgraphs that faithfully preserve the distances in the original graph up to small stretch. Spanner have been studied extensively as they have a wide range of applications ranging from distance oracles, labeling schemes and routing to solving linear systems and spectral sparsification. A k-spanner maintains pairwise distances up to multiplicative factor of k. It is a folklore that for every n-vertex graph G, one can construct a (2k − 1) spanner with O(n ^{1+1/k} ) edges. In a distributed setting, such spanners can be constructed in the standard CONGEST model using O(k ^{2} ) rounds, when randomization is allowed. In this work, we consider spanner constructions in the congested clique model, and show: a randomized construction of a (2k− 1)-spanner with O ^{e} (n ^{1+1/k} ) edges in O(log k) rounds. The previous best algorithm runs in O(k) rounds; a deterministic construction of a (2k−1)-spanner with O ^{e} (n ^{1+1/k} ) edges in O(log k+(log log n) ^{3} ) rounds. The previous best algorithm runs in O(k log n) rounds. This improvement is achieved by a new derandomization theorem for hitting sets which might be of independent interest; a deterministic construction of a O(k)-spanner with O(k · n ^{1+1/k} ) edges in O(log k) rounds.

Original language | English |
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Title of host publication | 32nd International Symposium on Distributed Computing, DISC 2018 |

Editors | Ulrich Schmid, Josef Widder |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770927 |

DOIs | |

State | Published - 1 Oct 2018 |

Externally published | Yes |

Event | 32nd International Symposium on Distributed Computing, DISC 2018 - New Orleans, United States Duration: 15 Oct 2018 → 19 Oct 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 121 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 32nd International Symposium on Distributed Computing, DISC 2018 |
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Country/Territory | United States |

City | New Orleans |

Period | 15/10/18 → 19/10/18 |

### Bibliographical note

Publisher Copyright:© Merav Parter and Eylon Yogev.

## Keywords

- And phrases Distributed Graph Algorithms
- Congested Clique
- Spanner