Advertisement

Time parameter and arbitrary deunions in the set union problem

  • Heikki Mannila
  • Esko Ukkonen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 318)

Abstract

The classical set union problem is to manipulate a partition of {1,2,...,n} under the operations find and union. We study two variants of this problem. In the first variant the find operations contain a time parameter. We show that this extended problem can be solved and requires ϑ(nlogn) time for n operations on separable pointer machines. In the second variant find operations are the usual ones, but an arbitrary union operation can be cancelled by the deunion operation. The same result holds for this variant. These problems are motivated by questions arising in the tracing of Prolog executions and in the incremental execution of logic programs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BT80]
    M.R. Brown and R.E. Tarjan: Design and analysis of a data structure for representing sorted lists. SIAM Journal on Computing 9, 3 (August 1980), 594–614.Google Scholar
  2. [DSST86]
    J.R. Driscoll, N. Sarnak, D.D. Sleator and R.E. Tarjan: Making data structures persistent. Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, 1986, p. 109–121.Google Scholar
  3. [vEB77]
    P. van Emde Boas: Preserving order in a forest in less than logarithmic time. Information Processing Letters 6, 3 (1977), 80–82.Google Scholar
  4. [GT83]
    H.N. Gabow and R.E. Tarjan: A linear-time algorithm for a special case of disjoint set union. Proceedings of the Fifteenth ACM SIGACT Symposium on Theory of Computing, 1983, p. 246–251.Google Scholar
  5. [Ho84]
    C.J. Hogger: Introduction to Logic Programming. Academic Press, 1984.Google Scholar
  6. [HM82]
    S. Huddleston and K. Mehlhorn: A new data structure for representing sorted lists. Acta Informatica 17 (1982), 157–184.Google Scholar
  7. [MU86a]
    H. Mannila and E. Ukkonen: On the complexity of unification sequences. Third International Conference on Logic Programming, E. Shapiro (ed.), Lecture Notes in Computer Science 225, Springer-Verlag 1986, p. 122–133.Google Scholar
  8. [MU86b]
    H. Mannila and E. Ukkonen: The set union problem with backtracking. Automata, Languages, and Programming, Thirteenth International Colloquium, L. Kott (ed.), Springer Verlag LNCS 226, 1986, 236–243.Google Scholar
  9. [MU86c]
    H. Manniala and E. Ukkonen: Timestamped term representation for implementing Prolog. Third IEEE Conference on Logic Programming, IEEE 1986, p. 159–167.Google Scholar
  10. [MU87a]
    H. Mannila and E. Ukkonen: Unifications, deunifications, and their complexity. Report C 1987–27, Department of Computer Science, University of Helsinki, 1987.Google Scholar
  11. [MU87b]
    H. Mannila and E. Ukkonen: Space-time optimal algorithms for the set union problem with backtracking. Report C-1987-80, Department of Computer Science, University of Helsinki, 1987.Google Scholar
  12. [MNA87]
    K. Mehlhorn, S. Näher, and H. Alt: A lower bound for the complexity of the union-splitfind problem. Automata, Languages, and Programming, Fourteenth International Colloquium, T. Ottmann (ed.), Springer-Verlag LNCS 267, 1987, p. 479–488.Google Scholar
  13. [Ov83]
    M.H. Overmars: The Design of Dynamic Data Structures. Lecture Notes in Computer Science 156, Springer-Verlag, 1983.Google Scholar
  14. [ST83]
    D.D. Sleator and R.E. Tarjan: A data structure for dynamic trees. Journal of Computer and System Sciences 26 (1983), 362–391.Google Scholar
  15. [ST85]
    D.D. Sleator and R.E. Tarjan: Self-adjusting binary search trees. Journal of the ACM 32, 3 (1985), 652–686.Google Scholar
  16. [Ta75]
    R.E. Tarjan: Efficiency of a good but not linear disjoint set union algorithm. Journal of the ACM 22, 2 (April 1975), 215–225.Google Scholar
  17. [Ta79]
    R.E. Tarjan: A class of algorithms which require nonlinear time to maintain disjoint sets. Journal of Computer and System Sciences 18 (1979), 110–127.Google Scholar
  18. [Ta85]
    R.E. Tarjan: Amortized computational complexity. SIAM Journal on Algebraic and Discrete Methods 6 (1985), 306–318.Google Scholar
  19. [TvL84]
    R.E. Tarjan and J. van Leeuwen: Worst-case analysis of set union algorithms. Journal of the ACM 31, 2 (April 1984), 245–281.Google Scholar
  20. [vLvW77]
    J. van Leeuwen and T. van der Weide: Alternative path compression techniques. Technical Report RUU-CS-77-3, Rijksuniversiteit Utrecht, Utrecht, The Netherlands.Google Scholar
  21. [WP77]
    D.H.D. Warren and L.M. Pereira: Prolog — the language and its implementation compared with LISP. ACM SIGPLAN Notices 12, 8 (August 1977), 109–115.Google Scholar
  22. [WT87]
    J. Westbrook and R.E. Tarjan: Amortized analysis of algorithms for set union with backtracking. Manuscript, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Heikki Mannila
    • 1
  • Esko Ukkonen
    • 1
  1. 1.Department of Computer ScienceUniversity of HelsinkiHelsinkiFinland

Personalised recommendations