## Abstract

We study the s-sources almost shortest paths (shortly, s-ASP) problem. Given an unweighted graph G = (V, E), and a subset S ⊆ V of s nodes, the goal is to compute almost shortest paths between all the pairs of nodes S × V. We devise an algorithm with running time O(|E|n^{ρ} + s · n^{1+ζ}) for this problem that computes the paths P_{u,w} for all pairs (u, w) ε S × V such that the length of P_{u,w} is at most (1 + ε)d_{G}(u, w) + β(ζ, ρ, ε) is constant when ζ, ρ and ε are (one can choose arbitrarily small constants ζ, ρ and ε). We also devise a distributed protocol for the s-ASP problem that computes the paths P_{u,w} as above, and has time and communication complexities of O(s · Diam(G)+n^{1+ζ/2}) (resp., O(s · Diam(G) log^{3} n + n^{1+ζ/2} log n)) and O(|E|n^{ρ} + s · n^{1+ζ}) (resp., O(|E|n^{ρ} + s · n^{1+ζ} + n^{1+ρ+ζ(ρ-ζ/2)/2})) in the synchronous (resp., asynchronous) setting. Our sequential algorithm, as well as the distributed protocol, is based on a novel algorithm for constructing (1 + ε, β(ζ, ρ, ε))-spanners of size O(n^{1+δ}), developed in this paper. This algorithm has running time of O(|E|n^{ρ}), which is significantly faster than the previously known algorithm of [20], whose running time is Õ(n^{2+ρ}). We also develop the first distributed protocol for constructing (1 + ε, β)-spanners. The time and communication complexities of this protocol are near-optimal.

Original language | English |
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Pages | 53-62 |

Number of pages | 10 |

DOIs | |

State | Published - 1 Jan 2001 |

Externally published | Yes |

Event | 20th Annual ACM Symposium on Principles of Distributed Computing - Newport, Rhode Island, United States Duration: 26 Aug 2001 → 29 Aug 2001 |

### Conference

Conference | 20th Annual ACM Symposium on Principles of Distributed Computing |
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Country/Territory | United States |

City | Newport, Rhode Island |

Period | 26/08/01 → 29/08/01 |

## Keywords

- Approximation algorithms
- Distributed computing

## ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications