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Succinct Orthogonal Range Search Structures on a Grid with Applications to Text Indexing

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Algorithms and Data Structures (WADS 2009)

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Abstract

We present a succinct representation of a set of n points on an n×n grid using \(n\lg n + o(n\lg n)\) bits to support orthogonal range counting in \(O(\lg n /\lg\lg n)\) time, and range reporting in \(O(k\lg n/\lg\lg n)\) time, where k is the size of the output. This achieves an improvement on query time by a factor of \(\lg\lg n\) upon the previous result of Mäkinen and Navarro [1], while using essentially the information-theoretic minimum space. Our data structure not only can be used as a key component in solutions to the general orthogonal range search problem to save storage cost, but also has applications in text indexing. In particular, we apply it to improve two previous space-efficient text indexes that support substring search [2] and position-restricted substring search [1]. We also use it to extend previous results on succinct representations of sequences of small integers, and to design succinct data structures supporting certain types of orthogonal range query in the plane.

This work was supported by NSERC of Canada. The work was done when the second author was in School of Computer Science, Carleton University, Canada.

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References

  1. Mäkinen, V., Navarro, G.: Rank and select revisited and extended. Theor. Comput. Sci. 387, 332–347 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chien, Y.F., Hon, W.K., Shah, R., Vitter, J.S.: Geometric burrows-wheeler transform: Linking range searching and text indexing. In: DCC, pp. 252–261 (2008)

    Google Scholar 

  3. Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: STOC, pp. 135–143 (1984)

    Google Scholar 

  4. Chazelle, B.: A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing 17(3), 427–462 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Overmars, M.H.: Efficient data structures for range searching on a grid. Journal of Algorithms 9(2), 254–275 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alstrup, S., Brodal, G.S., Rauhe, T.: New data structures for orthogonal range searching. In: FOCS, pp. 198–207 (2000)

    Google Scholar 

  7. Nekrich, Y.: Orthogonal range searching in linear and almost-linear space. Computational Geometry: Theory and Applications 42(4), 342–351 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jacobson, G.: Space-efficient static trees and graphs. In: FOCS, pp. 549–554 (1989)

    Google Scholar 

  9. Raman, R., Raman, V., Satti, S.R.: Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Transactions on Algorithms 3(4), 43 (2007)

    Article  MathSciNet  Google Scholar 

  10. Grossi, R., Gupta, A., Vitter, J.S.: High-order entropy-compressed text indexes. In: SODA, pp. 841–850 (2003)

    Google Scholar 

  11. Barbay, J., Golynski, A., Munro, J.I., Rao, S.S.: Adaptive searching in succinctly encoded binary relations and tree-structured documents. Theoretical Computer Science 387(3), 284–297 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Barbay, J., He, M., Munro, J.I., Rao, S.S.: Succinct indexes for strings, binary relations and multi-labeled trees. In: SODA, pp. 680–689 (2007)

    Google Scholar 

  13. Geary, R.F., Raman, R., Raman, V.: Succinct ordinal trees with level-ancestor queries. ACM Transactions on Algorithms 2(4), 510–534 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ferragina, P., Luccio, F., Manzini, G., Muthukrishnan, S.: Structuring labeled trees for optimal succinctness, and beyond. In: FOCS, pp. 184–196 (2005)

    Google Scholar 

  15. Ferragina, P., Manzini, G., Mäkinen, V., Navarro, G.: Compressed representations of sequences and full-text indexes. ACM Trans. Alg. 3(2), 20 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bauernöppel, F., Kranakis, E., Krizanc, D., Maheshwari, A., Sack, J.R., Urrutia, J.: Planar stage graphs: Characterizations and applications. Theoretical Computer Science 175(2), 239–255 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Clark, D.R., Munro, J.I.: Efficient suffix trees on secondary storage. In: SODA, pp. 383–391 (1996)

    Google Scholar 

  18. Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

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Bose, P., He, M., Maheshwari, A., Morin, P. (2009). Succinct Orthogonal Range Search Structures on a Grid with Applications to Text Indexing. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_9

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  • DOI: https://doi.org/10.1007/978-3-642-03367-4_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03366-7

  • Online ISBN: 978-3-642-03367-4

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