Advertisement

An Improved Succinct Representation for Dynamic k-ary Trees

  • Diego Arroyuelo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5029)

Abstract

k-ary trees are a fundamental data structure in many text-processing algorithms (e.g., text searching). The traditional pointer-based representation of trees is space consuming, and hence only relatively small trees can be kept in main memory. Nowadays, however, many applications need to store a huge amount of information. In this paper we present a succinct representation for dynamic k-ary trees of n nodes, requiring 2n + nlogk + o(nlogk) bits of space, which is close to the information-theoretic lower bound. Unlike alternative representations where the operations on the tree can be usually computed in O(logn) time, our data structure is able to take advantage of asymptotically smaller values of k, supporting the basic operations parent and child in O(logk + loglogn) time, which is o(logn) time whenever logk = o(logn). Insertions and deletions of leaves in the tree are supported in \(O((\log{k}+\log\log{n})(1+\frac{\log{k}}{\log{(\log{k} + \log\log{n})}}))\) amortized time. Our representation also supports more specialized operations (like subtreesize, depth, etc.), and provides a new trade-off when k = O(1) allowing faster updates (in O(loglogn) amortized time, versus the amortized time of O((loglogn)1 + ε ), for ε> 0, from Raman and Rao [21]), at the cost of slower basic operations (in O(loglogn) time, versus O(1) time of [21]).

Keywords

Candidate Node Extendible Array Parent Block Operation Insert Dynamic Data Structure 
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.
    Apostolico, A.: The myriad virtues of subword trees. In: Combinatorial Algorithms on Words. NATO ISI Series, pp. 85–96. Springer, Heidelberg (1985)Google Scholar
  2. 2.
    Arroyuelo, D., Navarro, G.: A Lempel-Ziv text index on secondary storage. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 83–94. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Arroyuelo, D., Navarro, G.: Space-efficient construction of LZ-index. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1143–1152. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Barbay, J., He, M., Munro, J.I., Rao, S.S.: Succinct indexes for strings, binary relations and multi-labeled trees. In: Proc. SODA, pp. 680–689 (2007)Google Scholar
  5. 5.
    Benoit, D., Demaine, E., Munro, J.I., Raman, R., Raman, V., Rao, S.S.: Representing trees of higher degree. Algorithmica 43(4), 275–292 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brodnik, A., Carlsson, S., Demaine, E., Munro, J.I., Sedgewick, R.: Resizable arrays in optimal time and space. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 37–48. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Chan, H.L., Hon, W.K., Lam, T.W., Sadakane, K.: Compressed indexes for dynamic text collections. ACM TALG 3(2) (article 21) (2007)Google Scholar
  8. 8.
    Ferragina, P., Luccio, F., Manzini, G., Muthukrishnan, S.: Structuring labeled trees for optimal succinctness, and beyond. In: Proc. FOCS, pp. 184–196 (2005)Google Scholar
  9. 9.
    Ferragina, P., Manzini, G., Mäkinen, V., Navarro, G.: Compressed representations of sequences and full-text indexes. ACM TALG 3(2) (article 20) (2007)Google Scholar
  10. 10.
    Geary, R., Raman, R., Raman, V.: Succinct ordinal trees with level-ancestor queries. In: Proc. SODA, pp. 1–10 (2004)Google Scholar
  11. 11.
    González, R., Navarro, G.: Improved dynamic rank-select entropy-bound structures. In: Proc. LATIN (to appear, 2008)Google Scholar
  12. 12.
    Hon, W.K., Sadakane, K., Sung, W.K.: Succinct data structures for searchable partial sums. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 505–516. Springer, Heidelberg (2003)Google Scholar
  13. 13.
    Jacobson, G.: Space-efficient static trees and graphs. In: Proc. FOCS, pp. 549–554 (1989)Google Scholar
  14. 14.
    Jansson, J., Sadakane, K., Sung, W.K.: Ultra-succinct representation of ordered trees. In: Proc. SODA, pp. 575–584 (2007)Google Scholar
  15. 15.
    Katajainen, J., Mäkinen, E.: Tree compression and optimization with applications. Int. J. Found. Comput. Sci. 1(4), 425–448 (1990)zbMATHCrossRefGoogle Scholar
  16. 16.
    Mäkinen, V., Navarro, G.: Dynamic entropy-compressed sequences and full-text indexes. ACM TALG (to appear, 2007)Google Scholar
  17. 17.
    Manzini, G.: An analysis of the Burrows-Wheeler transform. Journal of the ACM 48(3), 407–430 (2001)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Munro, J.I., Raman, V.: Succinct representation of balanced parentheses and static trees. SIAM Journal on Computing 31(3), 762–776 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Munro, J.I., Raman, V., Storm, A.: Representing dynamic binary trees succinctly. In: Proc. SODA, pp. 529–536 (2001)Google Scholar
  20. 20.
    Raman, R., Raman, V., Rao, S.S.: Succinct indexable dictionaries with applications to encoding k-ary trees and multisets. In: Proc. SODA, pp. 233–242 (2002)Google Scholar
  21. 21.
    Raman, R., Rao, S.S.: Succinct dynamic dictionaries and trees. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 357–368. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Diego Arroyuelo
    • 1
  1. 1.Dept. of Computer ScienceUniversity of Chile 

Personalised recommendations