Dynamic Algorithms for Graph Spanners

  • Surender Baswana
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)


Let G=(V,E) be an undirected weighted graph on |V|=n vertices and |E|=m edges. For the graph G, A spanner with stretch t∈ℕ is a subgraph (V,E S ), E S  ⊆ E, such that the distance between any pair of vertices in this subgraph is at most t times the distance between them in the graph G. We present simple and efficient dynamic algorithms for maintaining spanners with essentially optimal (expected) size versus stretch trade-off for any given unweighted graph. The main result is a decremental algorithm that takes expected \(O(\mathop{\mathrm{polylog}} n)\) time per edge deletion for maintaining a spanner with arbitrary stretch. This algorithm easily leads to a fully dynamic algorithm with sublinear (in n) time per edge insertion or deletion. Quite interestingly, this paper also reports that for stretch at most 6, it is possible to maintain a spanner fully dynamically with expected constant time per update. All these algorithms use simple randomization techniques on the top of an existing static algorithm [6] for computing spanners, and achieve drastic improvement over the previous best deterministic dynamic algorithms for spanners.


Hash Table Weighted Graph Dynamic Algorithm Edge Deletion Unweighted Graph 


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  1. 1.
    Althöfer, I., Das, G., Dobkin, D.P., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete and Computational Geometry 9, 81–100 (1993)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ausiello, G., Franciosa, P.G., Italiano, G.F.: Small stretch spanners on dynamic graphs. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 532–543. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Awerbuch, B.: Complexity of network synchronization. Journal of Ass. Compt. Mach. 804–823 (1985)Google Scholar
  4. 4.
    Awerbuch, B., Berger, B., Cowen, L., Peleg, D.: Near-linear time construction of sparse neighborhod covers. SIAM Journal on Computing 28, 263–277 (1998)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Baswana, S., Sen, S.: A simple linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Structures and Algorithms (to appear)Google Scholar
  6. 6.
    Baswana, S., Sen, S.: A simple linear time algorithm for computing a (2k − 1)-spanner of O(n 1 + 1/k) size in weighted graphs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 384–396. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in Õ(n2) time. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 271–280 (2004)Google Scholar
  8. 8.
    Bollobás, B.: Extremal Graph Theory. Academic Press, London (1978)MATHGoogle Scholar
  9. 9.
    Bondy, J.A., Simonovits, M.: Cycles of even length in graphs. Journal of Combinatorial Theory, Series B 16, 97–105 (1974)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cohen, E.: Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing 28, 210–236 (1998)MATHCrossRefGoogle Scholar
  11. 11.
    Erdős, P.: Extremal problems in graph theory. In: Theory of Graphs and its Applications (Proc. Sympos. Smolenice,1963), pp. 29–36. House Czechoslovak Acad. Sci., Prague (1964)Google Scholar
  12. 12.
    Even, S., Shiloach, Y.: An on-line edge-deletion problem. Journal of association for computing machinery 28, 1–4 (1981)MATHMathSciNetGoogle Scholar
  13. 13.
    Halperin, S., Zwick, U.: Linear time deterministic algorithm for computing spanners for unweighted graphs (unpublished manuscript, 1996)Google Scholar
  14. 14.
    Pagh, R., Rodler, F.F.: Cuckoo hashing. Journal of Algorithms 51, 122–144 (2004)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Peleg, D., Schäffer, A.: Graph spanners. Journal of Graph Theory 13, 99–116 (1989)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM Journal on Computing 18, 740–747 (1989)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. Journal of Assoc. Comp. Mach. 36(3), 510–530 (1989)MATHMathSciNetGoogle Scholar
  18. 18.
    Roditty, L., Thorup, M., Zwick, U.: Deterministic construction of approximate distance oracles and spanners. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 261–272. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Roditty, L., Zwick, U.: On dynamic shortest paths problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 580–591. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. 20.
    Thorup, M., Zwick, U.: Approximate distance oracles. Journal of Association of Computing Machinery 52, 1–24 (2005)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Surender Baswana
    • 1
  1. 1.Max-Planck Institute for Computer ScienceSaarbrückenGermany

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