Skip to main content
SpringerLink
Log in

Two-tier relaxed heaps

  • Original Article
  • Published:
Acta Informatica Aims and scope Submit manuscript

Abstract

We introduce a data structure which provides efficient heap operations with respect to the number of element comparisons performed. Let n denote the size of the heap being manipulated. Our data structure guarantees the worst-case cost of O(1) for finding the minimum, inserting an element, extracting an (unspecified) element, and replacing an element with a smaller element; and the worst-case cost of O(lg n) with at most lg n + 3 lg lg n + O(1) element comparisons for deleting an element. We thereby improve the comparison complexity of heap operations known for run-relaxed heaps and other worst-case efficient heaps. Furthermore, our data structure supports melding of two heaps of size m and n at the worst-case cost of O(min {lg m, lg n}).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brodal, G.S.: Worst-case efficient priority queues. In: Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58. ACM/SIAM, New York (1996)

  2. Brodal, G.S., Okasaki, C.: Optimal purely functional priority queues. J. Funct. Programming 6(6), 839–857 (1996)

    Article  MATH  Google Scholar 

  3. Clancy, M.J., Knuth, D.E.: A programming and problem-solving seminar. Technical Report STAN-CS-77-606, Department of Computer Science, Stanford University (1977)

  4. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press, Cambridge (2001)

    MATH  Google Scholar 

  5. Driscoll, J.R., Gabow, H.N., Shrairman, R., Tarjan, R.E.: Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation. Comm. ACM 31(11), 1343–1354 (1988)

    Article  MathSciNet  Google Scholar 

  6. Elmasry, A., Jensen, C., Katajainen, J.: A framework for speeding up priority-queue operations. CPH STL Report 2004-3, Department of Computing, University of Copenhagen. Available at http://cphstl.dk (2004)

  7. Elmasry, A., Jensen, C., Katajainen, J.: Two new methods for transforming priority queues into double-ended priority queues. CPH STL Report 2006-9, Department of Computing, University of Copenhagen Available at http://cphstl.dk (2006)

  8. Elmasry, A., Jensen, C., Katajainen, J.: Two-tier relaxed heaps. In: Proceedings of the 17th International Symposium on Algorithms and Computation. Lecture Notes in Computer Science, vol. 4288, pp. 308–317. Springer, Berlin (2006)

  9. Elmasry, A., Jensen, C., Katajainen, J.: Multipartite priority queues. ACM Transactions on Algorithms (2008, in press)

  10. Fredman, M.L.: On the efficiency of pairing heaps and related data structures. J. ACM 46(4), 473–501 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fredman, M.L., Sedgewick, R., Sleator, D.D., Tarjan, R.E.: The pairing heap: a new form of self-adjusting heap. Algorithmica 1(1), 111–129 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)

    Article  MathSciNet  Google Scholar 

  13. Gonnet, G.H., Munro, J.I.: Heaps on heaps. SIAM J. Comput. 15(4), 964–971 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Goodrich, M.T., Tamassia, R.: Algorithm Design: Foundations, Analysis, and Internet Examples. Wiley, New York (2002)

    MATH  Google Scholar 

  15. Kaplan, H., Shafrir, N., Tarjan, R.E.: Meldable heaps and Boolean union-find. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 573–582. ACM, New York (2002)

  16. Kaplan, H., Tarjan, R.E.: New heap data structures. Technical Report TR-597-99, Department of Computer Science, Princeton University (1999)

  17. Kernighan, B.W., Ritchie, D.M.: The C Programming Language, 2nd edn. Prentice Hall PTR, Englewood Cliffs (1988)

    Google Scholar 

  18. Okasaki, C.: Purely Functional Data Structures. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  19. Overmars, M.H., van Leeuwen, J.: Worst-case optimal insertion and deletion methods for decomposable searching problems. Inform. Process. Lett. 12(4), 168–173 (1981)

    Article  MATH  Google Scholar 

  20. Vuillemin, J.: A data structure for manipulating priority queues. Comm. ACM 21(4), 309–315 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  21. Williams, J.W.J.: Algorithm 232: Heapsort. Comm. ACM 7(6), 347–348 (1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Engineering and Systems, Alexandria University, Alexandria, Egypt

    Amr Elmasry

  2. Department of Computing, University of Copenhagen, Universitetsparken 1, 2100, Copenhagen East, Denmark

    Claus Jensen & Jyrki Katajainen

Authors
  1. Amr Elmasry
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claus Jensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jyrki Katajainen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jyrki Katajainen.

Additional information

A preliminary version of this paper [8] was presented at the 17th International Symposium on Algorithms and Computation held in Kolkata in December 2006.

Partially supported by the Danish Natural Science Research Council under contracts 21-02-0501 (project Practical data structures and algorithms) and 272-05-0272 (project Generic programming—algorithms and tools).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Elmasry, A., Jensen, C. & Katajainen, J. Two-tier relaxed heaps. Acta Informatica 45, 193–210 (2008). https://doi.org/10.1007/s00236-008-0070-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00236-008-0070-7

Keywords

Search

Navigation