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Deterministic Constructions of Approximate Distance Oracles and Spanners

  • Liam Roditty
  • Mikkel Thorup
  • Uri Zwick
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3580)

Abstract

Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in Õ(kmn \(^{\rm 1/{\it k}}\)) expected time and construct a data structure (a (2k –1)-approximate distance oracle) of size O(kn \(^{\rm 1+1/{\it k}}\)) capable of returning in O(k) time an approximation \(\hat{\delta}(u,v)\) of the distance δ(u,v) from u to v in G that satisfies \(\delta(u,v) \leq \hat{\delta}(u,v) \leq (2k -1)\cdot \delta(u,v)\), for any two vertices u,vV. They also presented a much slower Õ(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn \(^{\rm 1+1/{\it k}}\)log n). We present here a deterministic Õ(kmn \(^{\rm 1/{\it k}}\)) time algorithm for constructing oracles of size O(kn \(^{\rm 1+1/{\it k}}\)). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor.

Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn \(^{\rm 1+1/{\it k}}\)) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.

Keywords

Short Path Conditional Expectation Deterministic Algorithm Linear Time Algorithm Short Path Tree 
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.

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References

  1. 1.
    Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM Journal on Computing 28, 1167–1181 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Naor, M.: Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions. Algorithmica 16, 434–449 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Spencer, J.H.: The probabilistic method, 2nd edn. Wiley-Interscience, Hoboken (2000)zbMATHCrossRefGoogle Scholar
  4. 4.
    Awerbuch, B., Berger, B., Cowen, L., Peleg, D.: Near-linear time construction of sparse neighborhood covers. SIAM Journal on Computing 28, 263–277 (1999)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Baswana, S., Sen, S.: A simple linear time algorithm for computing (2k − 1)-spanner of O(n 1 + 1/k) size for weighted graphs. In: Proc. of 30th ICALP, pp. 384–296 (2003)Google Scholar
  6. 6.
    Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in O(n 2 logn) time. In: Proc. of 15th SODA, pp. 264–273 (2004)Google Scholar
  7. 7.
    Cohen, E.: Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing 28, 210–236 (1999)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Cohen, E., Zwick, U.: All-pairs small-stretch paths. Journal of Algorithms 38, 335–353 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms, 2nd edn. The MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  10. 10.
    Dor, D., Halperin, S., Zwick, U.: All pairs almost shortest paths. SIAM Journal on Computing 29, 1740–1759 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Elkin, M.: Computing almost shortest paths. In: Proc. of 20th PODC, pp. 53–62 (2001)Google Scholar
  12. 12.
    Elkin, M.L., Peleg, D.: (1 + ε,β)-Spanner constructions for general graphs. SIAM Journal on Computing 33(3), 608–631 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with O(1) worst case access time. Journal of the ACM 31, 538–544 (1984)zbMATHCrossRefGoogle Scholar
  14. 14.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM 34, 596–615 (1987)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Thorup, M.: Undirected single-source shortest paths with positive integer weights in linear time. Journal of the ACM 46, 362–394 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Thorup, M., Zwick, U.: Approximate distance oracles. Journal of the ACM 52(1), 1–24 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Liam Roditty
    • 1
  • Mikkel Thorup
    • 2
  • Uri Zwick
    • 1
  1. 1.School of Computer ScienceTel Aviv UniversityIsrael
  2. 2.AT&T Research LabsUSA

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