## Abstract

We study the s-sources almost shortest paths (abbreviated 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) ∈ S ×V such that the length of P_{u, w} is at most (1 + ϵ)d_{G}(u, w) + β(ζ, ρ, ϵ), and β(ζ, ρ, ϵ) is constant when ζ, ρ, and ϵ are arbitrarily small constants. 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}) (respectively, O(s. Diam(G) log^{3} n + n^{1+ ζ/2} log n)) and O(|E|n ^{ρ} + s. n^{1+ ζ}) (respectively, O(|E|n^{ρ} + s. n^{1+ ζ} + n^{1+ ρ+ ζ (ρ. ζ/2)/2})) in the synchronous (respectively 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+ ζ}), developed in this article. This algorithm has running time of O(|E|n ^{ρ}), which is significantly faster than the previously known algorithm given in Elkin and Peleg [2001], whose running time is.O(n^{2+ ρ}).We also develop the first distributed protocol for constructing (1+ϵ, β)-spanners. The communication complexity of this protocol is near optimal.

Original language | English |
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Pages (from-to) | 283-323 |

Number of pages | 41 |

Journal | ACM Transactions on Algorithms |

Volume | 1 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2005 |

## Keywords

- Algorithms
- Almost shortest paths
- Graph algorithms
- Spanners
- Theory