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Discrete & Computational Geometry

, Volume 9, Issue 1, pp 81–100 | Cite as

On sparse spanners of weighted graphs

  • Ingo Althöfer
  • Gautam Das
  • David Dobkin
  • Deborah Joseph
  • José Soares
Article

Abstract

Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v ɛV, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

Keywords

Planar Graph Minimum Span Tree Edge Weight Weight Graph Delaunay Triangulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [A]
    I. Althöfer, On Optimal Realizations of Finite Metric Spaces by Graphs,Discrete Comput. Geom. 3 (1988), 103–122.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Aw]
    B. Awerbuch, Complexity of Network Synchronization,J. Assoc. Comput. Mach. (1985), 804–823.Google Scholar
  3. [ABLP]
    B. Awerbuch, D. Bar-Noy, N. Linial, D. Peleg, Compact Distributed Data Structures for Adaptive Routing,Proc. STOC, 1989, pp. 479–489.Google Scholar
  4. [AP]
    B. Awerbuch, D. Peleg, Routing with Polynomial Communication-Space Tradeoff,SIAM J. Discrete Math., to appear.Google Scholar
  5. [BD]
    H. J. Bandelt, A. W. M. Dress, Reconstructing the Shape of a Tree from Observed Dissimilarity Data,Adv. in Appl. Math. 7 (1986), 309–343.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Be]
    M. Bern, Private Communication to David Dobkin, 1989.Google Scholar
  7. [Bo]
    B. Bollobás,Extremal Graph Theory, Academic Press, New York, 1978.zbMATHGoogle Scholar
  8. [C]
    L. P. Chew, There is a Planar Graph Almost as Good as the Complete Graph,Proc. ACM Symp. on Computational Geometry, 1986, pp. 169–177.Google Scholar
  9. [CS]
    J. H. Conway, N. J. A. Sloane,Sphere Packing, Lattices, and Groups, Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
  10. [D]
    A. W. M. Dress, Trees, Tight Extensions of Metric Spaces,Adv. in Math. 53 (1984), 321–402.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [DFS]
    D. P. Dobkin, S. J. Friedman, K. J. Supowit, Delaunay Graphs Are Almost as Good as Complete Graphs,Proc. FOCS, 1987, pp. 20–26.Google Scholar
  12. [DJ]
    G. Das, D. A. Joseph, Which Triangulations Approximate the Complete Graph?,Proc. Internat. Symp. on Optimal Algorithms, 1989, pp. 168–192, LNCS, Vol. 401, Springer-Verlag, Berlin.Google Scholar
  13. [ES]
    P. Erdös, H. Sachs, Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl,Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 12 (1963), 251–257.MathSciNetzbMATHGoogle Scholar
  14. [K]
    J. M. Keil, Approximating the Complete Euclidean Graph,Proc. SWAT, 1988, pp. 208–213, LNCS, Vol. 318, Springer-Verlag, Berlin.CrossRefGoogle Scholar
  15. [KG]
    J. M. Keil, C. A. Gutwin, The Delaunay Triangulation Closely Approximates the Complete Euclidean Graph,Proc. WADS, 1989, pp. 47–56, LNCS, Vol. 382, Springer-Verlag, Berlin.zbMATHGoogle Scholar
  16. [LL]
    C. Levcopoulos, A. Lingas, There Are Planar Graphs Almost as Good as the Complete Graphs and as Short as Minimum Spanning Trees,Proc. Internat. Symp. on Optimal Algorithms, 1989, pp. 9–13, LNCS, Vol. 401, Springer-Verlag, Berlin.Google Scholar
  17. [L]
    J. Longyear, Regulard-valent Graphs of Girth 6 and 2(d*dd+1) Vertices,J. Combin. Theory 9 (1970), 420–422.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [LPS]
    A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan Graphs,Combinatorica 8(3) (1988), 261–277.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [M]
    G. A. Margulis, Explicit Group-Theoretical Constructions of Combinatorial Schemes and Their Application to the Design of Expanders and Concentrators,Problems Inform. Transmission 24(1) (1988), 39–46 (translated fromProblemy Peredachi Informatsii).MathSciNetzbMATHGoogle Scholar
  20. [PS]
    D. Peleg, A. Schäffer, Graph Spanners,J. Graph Theory 13(1) (1989), 99–116.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [PUl]
    D. Peleg, J. Ullman, An Optimal Synchronizer for the Hypercube,SIAM J. Comput. 18 (1989), 740–747.MathSciNetCrossRefGoogle Scholar
  22. [PU]
    P. Peleg, E. Upfal, A Tradeoff Between Space and Efficiency for Routing Tables,Proc. STOC, 1988, pp. 43–52.Google Scholar
  23. [RS]
    P. Raghaven, M. Snir, Memory Versus Randomness in Online Algorithms,Proc. ICALP, 1989, pp. 687–703, LNCS, Vol. 372, Springer-Verlag, Berlin.Google Scholar
  24. [SV]
    R. Sedgewick, J. S. Vitter, Shortest Paths in Euclidean Graphs,Algorithmica 1 (1986), 31–48.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [S]
    J. M. S. Simões-Pereira, A Note on the Tree Realizability of a Distance Matrix,J. Combin. Theory 6 (1969), 303–310.CrossRefzbMATHGoogle Scholar
  26. [T]
    R. E. Tarjan,Data Structures and Network Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Ingo Althöfer
    • 1
  • Gautam Das
    • 2
  • David Dobkin
    • 3
  • Deborah Joseph
    • 4
  • José Soares
    • 5
    • 6
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Mathematical Sciences DepartmentMemphis State UniversityMemphisUSA
  3. 3.Department of Computer SciencePrinceton UniversityPrincetonUSA
  4. 4.Department of Computer SciencesUniversity of WisconsinMadisonUSA
  5. 5.Universidade de São Paulo, IME-USP/MACSao PaoloBrazil
  6. 6.Department of Computer ScienceUniversity of ChicagoChicagoUSA

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