Linear-Space Data Structures for Range Frequency Queries on Arrays and Trees

  • Stephane Durocher
  • Rahul Shah
  • Matthew Skala
  • Sharma V. Thankachan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)

Abstract

We present O(n)-space data structures to support various range frequency queries on a given array A[0:n − 1] or tree T with n nodes. Given a query consisting of an arbitrary pair of pre-order rank indices (i,j), our data structures return a least frequent element, mode, or α-minority of the multiset of elements in the unique path with endpoints at indices i and j in A or T. We describe a data structure that supports range least frequent element queries on arrays in \(O(\sqrt{n / w})\) time, improving the \(\Theta(\sqrt{n})\) worst-case time required by the data structure of Chan et al. (SWAT 2012), where w ∈ Ω(logn) is the word size in bits. We describe a data structure that supports range mode queries on trees in \(O(\log\log n \sqrt{n / w})\) time, improving the \(\Theta(\sqrt{n} \log n)\) worst-case time required by the data structure of Krizanc et al. (ISAAC 2003). Finally, we describe a data structure that supports range α-minority queries on trees in O(α− 1 loglogn) time, where α ∈ [0,1] is specified at query time.

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References

  1. 1.
    Belazzougui, D., Botelho, F.C., Dietzfelbinger, M.: Hash, displace, and compress. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 682–693. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Belazzougui, D., Gagie, T., Navarro, G.: Better space bounds for parameterized range majority and minority. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 121–132. Springer, Heidelberg (2013)Google Scholar
  3. 3.
    Chan, T.M., Durocher, S., Larsen, K.G., Morrison, J., Wilkinson, B.T.: Linear-space data structures for range mode query in arrays. In: Proc. STACS, vol. 14, pp. 291–301 (2012)Google Scholar
  4. 4.
    Chan, T.M., Durocher, S., Larsen, K.G., Morrison, J., Wilkinson, B.T.: Linear-space data structures for range mode query in arrays. Theory Comp. Sys. (2013)Google Scholar
  5. 5.
    Chan, T.M., Durocher, S., Skala, M., Wilkinson, B.T.: Linear-space data structures for range minority query in arrays. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 295–306. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Chazelle, B.: Filtering search: A new approach to query-answering. SIAM J. Comp. 15(3), 703–724 (1986)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Durocher, S., He, M., Munro, J.I., Nicholson, P.K., Skala, M.: Range majority in constant time and linear space. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 244–255. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Durocher, S., He, M., Munro, J.I., Nicholson, P.K., Skala, M.: Range majority in constant time and linear space. Inf. & Comp. 222, 169–179 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gagie, T., He, M., Munro, J.I., Nicholson, P.K.: Finding frequent elements in compressed 2D arrays and strings. In: Grossi, R., Sebastiani, F., Silvestri, F. (eds.) SPIRE 2011. LNCS, vol. 7024, pp. 295–300. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Hagerup, T., Tholey, T.: Efficient minimal perfect hashing in nearly minimal space. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 317–326. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Krizanc, D., Morin, P., Smid, M.: Range mode and range median queries on lists and trees. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 517–526. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Krizanc, D., Morin, P., Smid, M.: Range mode and range median queries on lists and trees. Nordic J. Computing 12, 1–17 (2005)MathSciNetMATHGoogle Scholar
  13. 13.
    Sadakane, K., Navarro, G.: Fully-functional succinct trees. In: Proc. ACM-SIAM SODA, pp. 134–149 (2010)Google Scholar
  14. 14.
    Schmidt, J.P., Siegel, A.: The spatial complexity of oblivious k-probe hash functions. SIAM J. Comput. 19(5), 775–786 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Rahul Shah
    • 2
  • Matthew Skala
    • 1
  • Sharma V. Thankachan
    • 2
  1. 1.University of ManitobaWinnipegCanada
  2. 2.Louisiana State UniversityBaton RougeUSA

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