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Roundtrip spanners and roundtrip routing in directed graphs

Published: 04 July 2008 Publication History

Abstract

We introduce the notion of roundtrip-spanners of weighted directed graphs and describe efficient algorithms for their construction. We show that for every integer k ≥ 1 and any ϵ > 0, any directed graph on n vertices with edge weights in the range [1, W] has a (2k + ϵ)-roundtrip-spanner with O(min{(k2/ϵ) n1 + 1/k (log(nW), (k/ϵ)2 n1 + 1/k,(log n)2−1/k}) edges. We then extend these constructions and obtain compact roundtrip routing schemes. For every integer k ≥ 1 and every ϵ > 0, we describe a roundtrip routing scheme that has stretch 4k + ϵ, and uses at each vertex a routing table of size Õ((k2/ϵ)n1/k log(nW)). We also show that any weighted directed graph with arbitrary/ positive edge weights has a 3-roundtrip-spanner with O(n3/2) edges. This result is optimal. Finally, we present a stretch 3 roundtrip routing scheme that uses local routing tables of size Õ(n1/2). This routing scheme is essentially optimal. The roundtrip-spanner constructions and the roundtrip routing schemes for directed graphs that we describe are only slightly worse than the best available spanners and routing schemes for undirected graphs. Our roundtrip routing schemes substantially improve previous results of Cowen and Wagner. Our results are obtained by combining ideas of Cohen, Cowen and Wagner, Thorup and Zwick, with some new ideas.

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cover image ACM Transactions on Algorithms
ACM Transactions on Algorithms  Volume 4, Issue 3
June 2008
270 pages
ISSN:1549-6325
EISSN:1549-6333
DOI:10.1145/1367064
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 04 July 2008
Accepted: 01 March 2007
Received: 01 September 2006
Published in TALG Volume 4, Issue 3

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Author Tags

  1. Routing
  2. distances
  3. roundtrip
  4. shortest paths
  5. spanners

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