Advertisement

Representing Trees of Higher Degree

  • David Benoit
  • Erik D. Demaine
  • J. Ian Munro
  • Venkatesh Raman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1663)

Abstract

This paper focuses on space efficient representations of trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (rooted trees where each node has k positions each of which may have a reference to a child) and ordinal trees (the children of each node are simply ordered). Our representations use a number of bits within a lower order term of the information theoretic lower bound. For cardinal trees the structure supports finding the parent, child i or subtree size of a given node. For ordinal trees we support the operations of finding the degree, parent, ith child and subtree size. These operations provide a mapping from the n nodes of the tree onto the integers [1, n] and all are performed in constant time, except finding child i in cardinal trees. For k-ary cardinal trees, this operation takes O(lg lg k) time for the worst relationship between k and n, and constant time if k is much less than n.

Keywords

Binary Tree Suffix Tree Prefix Code Auxiliary Structure Balance String 
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.
    David Benoit. Compact Tree Representations. MMath thesis, U. Waterloo, 1998.Google Scholar
  2. 2.
    Andrej Brodnik and J. Ian Munro. Membership in constant time and almost minimum space. SIAM J. Computing, to appear.Google Scholar
  3. 3.
    David Clark. Compact Pat Trees. PhD thesis, U. Waterloo, 1996.Google Scholar
  4. 4.
    David R. Clark and J. Ian Munro. Efficient suffix trees on secondary storage. In Proc. ACM-SIAM Symposium on Discrete Algorithms, 383–391, 1996.Google Scholar
  5. 5.
    John J. Darragh, John G. Cleary, and Ian H. Whitten. Bonsai: a compact representation of trees. Software|Practice and Experience, 23(3):277–291, March 1993.CrossRefGoogle Scholar
  6. 6.
    A. Fiat and M. Naor. Implicit O(1) probe search. SIAM J. Computing, 22(1):1–10, January 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A. Fiat, M. Naor, J. P. Schmidt, and A. Siegel. Nonoblivious hashing. J. ACM, 39(4):764–782, April 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Michael L. Fredman, János Komlós, and Endre Szemerédi. Storing a sparse table with O(1) worst case access time. J. ACM, 31(3):538–544, July 1984.zbMATHCrossRefGoogle Scholar
  9. 9.
    Michael L. Fredman and Dan E. Willard. Surpassing the information theoretic bound with fusion trees. J. Computer and System Sciences, 47(3):424–436, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    G. H. Gonnet, R.A. Baeza-Yates, and T. Snider. New indicies for text: PAT trees and PAT arrays. In Information Retrieval: Data Structures & Algorithms, 66–82, Prentice Hall, 1992.Google Scholar
  11. 11.
    Guy Jacobson. Space-efficient static trees and graphs. In Proc. 30th Annual Symposium on Foundations of Computer Science, 549–554, 1989.Google Scholar
  12. 12.
    Guy Jacobson. Succinct Static Data Structures. PhD thesis, CMU, 1989.Google Scholar
  13. 13.
    Udi Manber and Gene Myers. Suffix arrays: A new method for on-line string searches. SIAM J. Computing, 22(5):935–948, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    J. Ian Munro. Tables. In Proc. 16th Conf. on the Foundations of Software Technology and Theoretical Computer Science, LNCS vol. 1180, 37–42, Springer, 1996.CrossRefGoogle Scholar
  15. 15.
    J. Ian Munro and Venkatesh Raman. Succinct representation of balanced parentheses, static trees and planar graphs. In Proc. 38th Annual Symposium on Foundations of Computer Science, 118–126, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • David Benoit
    • 1
    • 2
  • Erik D. Demaine
    • 2
  • J. Ian Munro
    • 2
  • Venkatesh Raman
    • 3
  1. 1.InfoInteractive Inc.BedfordCanada
  2. 2.Dept. of Computer ScienceUniversity of WaterlooWaterlooCanada
  3. 3.Institute of Mathematical SciencesChennaiIndia

Personalised recommendations