## Abstract

For an unweighted undirected graph G = (V, E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G′ = (V, H), H ⊆ E, is called an (α. β)-spanner of G if for every pair of vertices u, v ε V, dist_{G′}(u, v) ≤ α · dist _{G}(u, v) + β. It was shown in [21] that for any ε > 0, κ = 1, 2,..., there exists an integer β= β(ε, κ) such that for every n-vertex graph G there exists a (1 + ε, β)-spanner G′ with O(n^{1+1/κ}) edges. An efficient distributed protocol for constructing (1 + ε, β)-spanners was devised in [19]. The running time and the communication complexity of that protocol are O(n ^{1+ρ}) and O(|E|n^{ρ}), respectively, where ρ is an additional control parameter of the protocol that affects only the additive term β. In this paper we devise a protocol with a drastically improved running time (O(n^{ρ}) as opposed to O(n^{1+ρ})) for constructing (1 + ε, β)-spanners. Our protocol has the same communication complexity as the protocol of [19], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [19]. The protocol can be easily extended to a parallel implementation which runs in O(log n + (|E\ · n^{ρ} log n) / p) time using p processors in the EREW PRAM model. In particular, when the number of processors, p, is at least |E| · n^{ρ}, the running time of the algorithm is O(log n). We also show that our protocol for constructing (1 + ε, β)-spanners can be adapted to the streaming model, and devise a streaming algorithm that uses a constant number of passes and O (n^{1+1/κ}· log n) bits of space for computing all-pairs-almosl-shortest-paths of length at most by a multiplicative factor (1 + ε) and an additive term of β greater than the shortest paths. Our algorithm processes each edge in time O(n^{ρ}), for an arbitrarily small ρ > 0. The only previously known algorithm for the problem [23] constructs paths of length κ times greater than the shortest paths, has the same space requirements as our algorithm, but requires O(n ^{1+1/κ}) time for processing each edge of the input graph. However, the algorithm of [23] uses just one pass over the input, as opposed to the constant number of passes in our algorithm.^{1} We also show that any streaming algorithm for o(n)-approximate distance computation requires Ω(n) bits of space.

Original language | English |
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Pages (from-to) | 375-385 |

Number of pages | 11 |

Journal | Distributed Computing |

Volume | 18 |

Issue number | 5 |

DOIs | |

State | Published - 1 Apr 2006 |

Externally published | Yes |

## Keywords

- Almost shortest paths
- Spanner
- Streaming model

## ASJC Scopus subject areas

- Theoretical Computer Science
- Hardware and Architecture
- Computer Networks and Communications
- Computational Theory and Mathematics