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A Uniform Approach Towards Succinct Representation of Trees

  • Arash Farzan
  • J. Ian Munro
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5124)

Abstract

We propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decomposition of trees into subtrees. Recursive decomposition of a structure into substructures is a common technique in succinct data structures and has been shown fruitful in succinct representation of ordinal trees [7,10] and dynamic binary trees [16,21]. We take an approach that simplifies the existing representation of ordinal trees while allowing the full set of navigational operations. The approach applied to cardinal (i.e. k-ary) trees yields a space-optimal succinct representation allowing cardinal-type operations (e.g. determining child labeled i) as well as the full set of ordinal-type operations (e.g. reporting the number of siblings to the left of a node). Existing space-optimal succinct representations had not been able to support both types of operations [2,19].

We demonstrate how the approach can be applied to obtain a space-optimal succinct representation for the family of free trees where the order of children is insignificant. Furthermore, we show that our approach can be used to obtain entropy-based succinct representations. We show that our approach matches the degree-distribution entropy suggested by Jansson et al. [13]. We discuss that our approach can be made adaptive to various other entropy measures.

Keywords

Lookup Table Free Tree Dummy Node Recursive Decomposition Succinct Representation 
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.

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References

  1. 1.
    Barbay, J., He, M., Munro, J.I., Rao, S.S.: Succinct indexes for strings, binary relations and multi-labeled trees. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 680–689. ACM-SIAM, New York (2007)Google Scholar
  2. 2.
    Benoit, D., Demaine, E.D., Munro, J.I., Raman, R., Raman, V., Rao, S.S.: Representing trees of higher degree. Algorithmica 43(4), 275–292 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bernhart, F.R.: Catalan, motzkin, and riordan numbers. Discrete Mathematics 204, 72–112 (1999)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Clark, D.R.: Compact pat trees. PhD thesis, Waterloo, Ontario. Canada (1998)Google Scholar
  5. 5.
    Clark, D.R., Munro, J.I.: Efficient suffix trees on secondary storage (extended abstract). In: SODA, pp. 383–391 (1996)Google Scholar
  6. 6.
    Etherington, I.M.H.: Non-associate powers and a functional equation. The Mathematical Gazette 21(242), 36–39 (1937)zbMATHCrossRefGoogle Scholar
  7. 7.
    Geary, R.F., Raman, R., Raman, V.: Succinct ordinal trees with level-ancestor queries. ACM Transactions on Algorithms 2(4), 510–534 (2006)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley Longman Publishing Co. Inc. Boston (1994)zbMATHGoogle Scholar
  9. 9.
    Harary, F., Palmer, E.M.: Graphical Enuemration. Academic Press, New York (1973)Google Scholar
  10. 10.
    He, M., Munro, J.I., Rao, S.S.: Succinct ordinal trees based on tree covering. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 509–520. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Jacobson, G.: Space-efficient static trees and graphs. In: Foundations of Computer Science. 30th Annual Symposium on (30 October-1 November 1989), pp. 549–554 (1989)Google Scholar
  12. 12.
    Jacobson, G.J. Succinct static data structures. PhD thesis, Pittsburgh, PA, USA (1988)Google Scholar
  13. 13.
    Jansson, J., Sadakane, K., Sung, W.-K.: Ultra-succinct representation of ordered trees. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 575–584. SIAM, Philadelphia (2007)Google Scholar
  14. 14.
    Knuth, D.E.: The Art of Computer Programming, 3rd edn. vol. 1. Addison-Wesley, Reading (1997)Google Scholar
  15. 15.
    Munro, J.I., Raman, V.: Succinct representation of balanced parentheses, static trees and planar graphs. In: IEEE Symposium on Foundations of Computer Science, pp. 118–126 (1997)Google Scholar
  16. 16.
    Munro, J.I., Raman, V., Storm, A.J.: Representing dynamic binary trees succinctly. In: SODA, pp. 529–536 (2001)Google Scholar
  17. 17.
    Odlyzko, A.M.: Some new methods and results in tree enumeration, (May 04, 1984)Google Scholar
  18. 18.
    Otter, R.: The number of trees. The Annals of Mathematics, 2nd Ser. 49(3), 583–599 (1948)MathSciNetGoogle Scholar
  19. 19.
    Raman, R., Raman, V., Rao, S.S.: Succinct indexable dictionaries with applications to encoding k-ary trees and multisets. In: SODA, pp. 233–242 (2002)Google Scholar
  20. 20.
    Rote, G.: Binary trees having a given number of nodes with 0,1, and 2 children. Séminaire Lotharingien de Combinatoire 38 (1997)Google Scholar
  21. 21.
    Storm, A.J. Representing dynamic binary trees succinctly. Master’s thesis, School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada (2000)Google Scholar
  22. 22.
    Wedderburn, J.H.M.: The Functional Equation g(x 2) = 2ax + [g(x)]2. The Annals of Mathematics, 2nd Ser. 24(2), 121–140 (1922)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Arash Farzan
    • 1
  • J. Ian Munro
    • 1
  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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