Advertisement

Stochastic graphs have short memory: Fully dynamic connectivity in poly-log expected time

  • S. Nikoletseas
  • J. Reif
  • P. Spirakis
  • M. Yung
Communication Protocols
Part of the Lecture Notes in Computer Science book series (LNCS, volume 944)

Abstract

This paper introduces average case analysis of fully dynamic graph connectivity (when the operations are edge insertions and deletions). To this end we introduce the model of stochastic graph processes, i.e. dynamically changing random graphs with random equiprobable edge insertions and deletions, which generalizes Erdös and Renyi's 35 year-old random graph process. As the stochastic graph process continues indefinitely, all potential edge locations (in V × V) may be repeatedly inspected (and learned) by the algorithm. This learning of the structure seems to imply that traditional random graph analysis methods cannot be employed (since an observed edge is not a random event anymore). However, we show that a small (logarithmic) number of dynamic random updates are enough to allow our algorithm to re-examine edges as if they were random with respect to certain events (i.e. the graph “forgets” its structure). This short memory property of the stochastic graph process enables us to present an algorithm for graph connectivity which admits an amortized expected cost of O(log3n) time per update. In contrast, the best known deterministic worst-case algorithms for fully dynamic connectivity require n1/2 time per update.

Keywords

Random Graph Neighborhood Search Giant Component Dynamic Algorithm Connectivity Query 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Alberts and M. Rauch, “Average Case Analysis of Dynamic Graph Algorithms”, 6th SODA, 1995.Google Scholar
  2. 2.
    D. Angluin and L. Valiant, “Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings”, JCSS, vol. 18, pp. 155–193, 1979.Google Scholar
  3. 3.
    B. Bollobás, “Random Graphs”, Academic Press, 1985.Google Scholar
  4. 4.
    J. Cheriyan and R. Thurimella, “Algorithms for parallel k-vertex connectivity and sparse certificates”, 23rd STOC, pp. 391–401, 1991.Google Scholar
  5. 5.
    D. Coppersmith, P. Raghavan and M. Tompa, “Parallel graph algorithms that are efficient on the average”, 28th FOCS, pp. 260–270, 1987.Google Scholar
  6. 6.
    D. Eppstein, G. Italiano, R. Tamassia, R. Tarjan, J. Westbrook and M. Yung, “Maintenance of a minimum spanning forest in a dynamic plane graph”, 1st SODA, pp. 1–11, 1990.Google Scholar
  7. 7.
    D. Eppstein, Z. Galil, G. Italiano and A. Nissenzweig, “Sparsification-A technique for speeding up Dynamic Graph Algorithms”, 33rd FOCS, 1992.Google Scholar
  8. 8.
    P. Erdös and A. Renyi, “On the evolution of random graphs”, Magyar Tud. Akad. Math. Kut. Int. Kozl. 5, pp. 17–61, 1960.Google Scholar
  9. 9.
    G. Frederikson, “Data structures for on-line updating of minimum spanning trees”, SIAM J. Comput., 14, pp. 781–798, 1985.Google Scholar
  10. 10.
    G. Frederikson, “Ambivalent data structures for dynamic 2-edge-connectivity and k-smallest spanning trees”, 32nd FOCS, pp. 632–641, 1991.Google Scholar
  11. 11.
    G. Frederikson, “A data structure for dynamically maintaining rooted trees”, 4th SODA, 1993.Google Scholar
  12. 12.
    A. Frieze, “Probabilistic Analysis of Graph Algorithms”, Feb. 1989.Google Scholar
  13. 13.
    Z. Galil and G. Italiano, “Fully dynamic algorithms for edge connectivity problems”, 23rd STOC, 1991.Google Scholar
  14. 14.
    Z. Galil, G. Italiano and N. Sarnak, “Fully Dynamic Planarity Testing”, 24th STOC, 1992.Google Scholar
  15. 15.
    M. Rauch Henzinger and V. King, “Randomized Dynamic Algorithms with Polylogarithmic Time per Operation”, 27-th STOC, 1995.Google Scholar
  16. 16.
    R. Karp, “Probabilistic Recurrence Relations”, 23rd STOC, pp. 190–197, 1991.Google Scholar
  17. 17.
    R. Karp and M. Sipser, “Maximum matching in sparse random graphs”, 22nd FOCS, pp. 364–375, 1981.Google Scholar
  18. 18.
    R. Karp and R. Tarjan, “Linear expected time for connectivity problems”, 12th STOC, 1980.Google Scholar
  19. 19.
    R. Motwani, “Expanding graphs and the average-case analysis of algorithms for matching and related problems”, 21st STOC, pp. 550–561, 1989.Google Scholar
  20. 20.
    S. Nikoletseas and P. Spirakis, “Expander Properties in Random Regular Graphs with Edge Faults”, 12th STACS, pp. 421–432, 1995.Google Scholar
  21. 21.
    S. Nikoletseas and P. Spirakis, “Near-Optimal Dominating Sets in Dense Random Graphs in Polynomial Expected Time”, 19th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), 1993.Google Scholar
  22. 22.
    S. Nikoletseas, K. Palem, P. Spirakis and M. Yung, “Short Vertex Disjoint Paths and Multiconnectivity in Random Graphs: Reliable Network Computing”, 21st ICALP, pp. 508–515, 1994.Google Scholar
  23. 23.
    S. Nikoletseas, J. Reif, P. Spirakis and M. Yung, “Stochastic Graphs Have Short Memory: Fully Dynamic Connectivity in Poly-Log Expected Time”, Technical Report T.R. 94.04.25, Computer Technology Institute (CTI), Patras, 1994.Google Scholar
  24. 24.
    J. Reif, “A topological approach to dynamic graph connectivity”, Inform. Process. Lett., 25, pp. 65–70, 1987.Google Scholar
  25. 25.
    J. Reif and P. Spirakis, “Expected parallel time analysis and sequential space complexity of graph and digraph problems”, Algorithmica, 1992.Google Scholar
  26. 26.
    D. Sleator and R. Tarjan, “A data structure for dynamic trees”, J. Comput. System Sci., 24, pp. 362–381, 1983.Google Scholar
  27. 27.
    J. Spencer, “Ten Lectures on the Probabilistic Method”, SIAM, 1987.Google Scholar
  28. 28.
    P. Spira and A. Pan, “On finding and updating spanning trees and shortest paths”, SIAM J. Comput., 4, pp. 375–380, 1975.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • S. Nikoletseas
    • 1
  • J. Reif
    • 2
  • P. Spirakis
    • 1
  • M. Yung
    • 3
  1. 1.Computer Technology InstitutePatrasGreece
  2. 2.Department of Computer ScienceDuke UniversityUK
  3. 3.IBM Research DivisionT. J. Watson Research CenterUSA

Personalised recommendations