Succinct Data Structures for Path Queries

  • Meng He
  • J. Ian Munro
  • Gelin Zhou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)


Consider a tree T on n nodes, each having a weight drawn from [1..σ]. In this paper, we design succinct data structures to encode T using \(n H(W_T) + o(n\lg \sigma)\) bits of space, such that we can support path counting queries in \(O(\frac{\lg \sigma}{\lg\lg n} + 1)\) time, path reporting queries in \(O((occ+1)(\frac{\lg \sigma}{\lg\lg n} + 1))\) time, and path median and path selection queries in \(O(\frac{\lg \sigma}{\lg\lg \sigma})\) time, where H(W T ) is the entropy of the multiset of the weights of the nodes in T. Our results not only improve the best known linear space data structures [15], but also match the lower bounds for these path queries [18,19,16] when \(\sigma = \Omega(n / \textrm{polylog}(n))\).


Query Time Path Query Space Cost Lower Common Ancestor Cover Element 
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  1. 1.
    Alon, N., Schieber, B.: Optimal preprocessing for answering on-line product queries. Tech. rep., Tel Aviv University (1987)Google Scholar
  2. 2.
    Barbay, J., Golynski, A., Munro, J.I., Rao, S.S.: Adaptive Searching in Succinctly Encoded Binary Relations and Tree-Structured Documents. In: Lewenstein, M., Valiente, G. (eds.) CPM 2006. LNCS, vol. 4009, pp. 24–35. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Barbay, J., He, M., Munro, J.I., Rao, S.S.: Succinct indexes for strings, binary relations and multilabeled trees. ACM Transactions on Algorithms 7(4), 52 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bille, P.: A survey on tree edit distance and related problems. Theor. Comput. Sci. 337(1-3), 217–239 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bose, P., He, M., Maheshwari, A., Morin, P.: Succinct Orthogonal Range Search Structures on a Grid with Applications to Text Indexing. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 98–109. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Brodal, G.S., Gfeller, B., Jørgensen, A.G., Sanders, P.: Towards optimal range medians. Theor. Comput. Sci. 412(24), 2588–2601 (2011)zbMATHCrossRefGoogle Scholar
  7. 7.
    Chazelle, B.: Computing on a free tree via complexity-preserving mappings. Algorithmica 2, 337–361 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Farzan, A., Munro, J.I.: A Uniform Approach Towards Succinct Representation of Trees. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 173–184. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Farzan, A., Raman, R., Rao, S.S.: Universal Succinct Representations of Trees? In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 451–462. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Ferragina, P., Luccio, F., Manzini, G., Muthukrishnan, S.: Compressing and indexing labeled trees, with applications. J. ACM 57(1) (2009)Google Scholar
  11. 11.
    Ferragina, P., Manzini, G., Mäkinen, V., Navarro, G.: Compressed representations of sequences and full-text indexes. ACM Transactions on Algorithms 3(2) (2007)Google Scholar
  12. 12.
    Geary, R.F., Raman, R., Raman, V.: Succinct ordinal trees with level-ancestor queries. ACM Transactions on Algorithms 2(4), 510–534 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hagerup, T.: Parallel preprocessing for path queries without concurrent reading. Inf. Comput. 158(1), 18–28 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    He, M., Munro, J.I., Zhou, G.: Path Queries in Weighted Trees. In: Asano, T., Nakano, S.-I., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 140–149. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Jørgensen, A.G., Larsen, K.G.: Range selection and median: Tight cell probe lower bounds and adaptive data structures. In: SODA, pp. 805–813 (2011)Google Scholar
  17. 17.
    Krizanc, D., Morin, P., Smid, M.H.M.: Range mode and range median queries on lists and trees. Nord. J. Comput. 12(1), 1–17 (2005)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pǎtraşcu, M.: Lower bounds for 2-dimensional range counting. In: STOC, pp. 40–46 (2007)Google Scholar
  19. 19.
    Pǎtraşcu, M.: Unifying the landscape of cell-probe lower bounds. SIAM J. Comput. 40(3), 827–847 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Raman, R., Raman, V., Rao, S.S.: Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Transactions on Algorithms 3(4) (2007)Google Scholar
  21. 21.
    Sadakane, K., Navarro, G.: Fully-functional succinct trees. In: SODA, pp. 134–149 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Meng He
    • 1
  • J. Ian Munro
    • 2
  • Gelin Zhou
    • 2
  1. 1.Faculty of Computer ScienceDalhousie UniversityCanada
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada

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