Efficient algorithms for constructing (1+6, β)-spanners in the distributed and streaming models

Michael Elkin, Jian Zhang

Research output: Contribution to conferencePaperpeer-review

24 Scopus citations

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 ∈ V, distcG' (u, v) ≤ α · distcG(u,v) + β. It was shown in [20] that for any ε > 0, K = 1, 2,., there exists an integer β= β(ε, K) such that for every n-vertex graph G there exists a (1+ ε, β)-spanner G' with O(n1+1/K) edges. An efficient distributed protocol for constructing (1 + ε, β)-spanners was devised in [18]. The running time and the communication complexity of that protocol are O(n1+p) and O(\E\nρ), respectively, where p 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(np) as opposed to O(n 1+p)) for constructing (1 + ε, β)-spanners. Our protocol has the same communication complexity as the protocol of [18], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [18], 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(n1+1/K · log n) bits of space for computing all-pairs-almost-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(np), for an arbitrarily small p > 0. The only previously known algorithm for the problem [21] constructs paths of length K times greater than the shortest paths, has the same space requirements as our algorithm, but requires O(n1+l/K) time for processing each edge of the input graph. However, the algorithm of [21] uses just one pass over the input, as opposed to the constant number of passes in our algorithm. We also show that any streaming algorithm for o(n) -approximate distance computation requires Ω(n) bits of space.

Original languageEnglish
Pages160-168
Number of pages9
StatePublished - 27 Dec 2004
Externally publishedYes
EventProceedings of the 23rd Annual ACM Symposium on Principles of Distributed Computing - St. John's, Nfld., Canada
Duration: 25 Jul 200428 Jul 2004

Conference

ConferenceProceedings of the 23rd Annual ACM Symposium on Principles of Distributed Computing
Country/TerritoryCanada
CitySt. John's, Nfld.
Period25/07/0428/07/04

Keywords

  • Almost shortest paths
  • Spanner
  • Streaming model

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