Skip to main content
SpringerLink
Log in

Linear-Space Data Structures for Range Mode Query in Arrays

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

A mode of a multiset S is an element a∈S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1:n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i,j) for which a mode of A[i:j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (Proceedings of the International Symposium on Algorithms and Computation (ISAAC), pp. 517–526, 2003), requires \(\varTheta (\sqrt{n}\log\log n)\) query time in the worst case. We improve their result and present an O(n)-space data structure that supports range mode queries in \(O(\sqrt{n/\log n})\) worst-case time. In the external memory model, we give a linear-space data structure that requires \(O(\sqrt{n/B})\) I/Os per query, where B denotes the block size. Furthermore, we present strong evidence that a query time significantly below \(\sqrt{n}\) cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two \(\sqrt{n} \times \sqrt{n}\) matrices reduces to n range mode queries in an array of size O(n).

Additionally, we give linear-space data structures for the dynamic problem (queries and updates in near O(n 3/4) time), for orthogonal range mode in d dimensions (queries in near O(n 1−1/2d) time) and for half-space range mode in d dimensions (queries in \(O(n^{1-1/d^{2}})\) time). Finally, we complement our dynamic data structure with a reduction from the multiphase problem, again supporting that we cannot hope for much more efficient data structures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. The original data structure described by Krizanc et al. [46] supports queries in \(O(\sqrt{n}\log n)\) time. As they remarked, this time can be reduced to \(O(\sqrt{n}\log\log n)\) by using van Emde Boas trees for predecessor search [58].

  2. Although the time required to complete a linear scan could be reduced by instead using a binary search or a more efficient predecessor data structure, the asymptotic worst-case time remains unchanged; for simplicity, a linear scan suffices.

  3. Succinct data structures can ensure that space usage is very close to the length of the bit string up to lower-order terms, but this fact is not needed in our application.

References

  1. Agarwal, P.K.: Range searching. In: Goodman, J., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 809–837. CRC Press, New York (2004)

    Google Scholar 

  2. Alon, N., Schieber, B.: Optimal preprocessing for answering on-line product queries. Technical report 71/87, Tel-Aviv University (1987)

  3. Amir, A., Fischer, J., Lewenstein, M.: Two-dimensional range minimum queries. In: Proceedings of the Symposium on Combinatorial Pattern Matching (CPM). Lecture Notes in Computer Science, vol. 4580, pp. 286–294. Springer, Berlin (2007)

    Chapter  Google Scholar 

  4. Bansal, N., Williams, R.: Regularity lemmas and combinatorial algorithms. In: Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pp. 745–754 (2009)

    Google Scholar 

  5. Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Proceedings of the Latin American Theoretical Informatics Symposium (LATIN). Lecture Notes in Computer Science, vol. 1776, pp. 88–94. Springer, Berlin (2000)

    Google Scholar 

  6. Berkman, O., Breslauer, D., Galil, Z., Schieber, B., Vishkin, U.: Highly parallelizable problems. In: Proceedings of the ACM Symposium on the Theory of Computing (STOC), pp. 309–319 (1989)

    Google Scholar 

  7. Bose, P., Kranakis, E., Morin, P., Tang, Y.: Approximate range mode and range median queries. In: Proceedings of the International Symposium on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Computer Science, vol. 3404, pp. 377–388. Springer, Berlin (2005)

    Google Scholar 

  8. Brodal, G.S., Jørgensen, A.G.: Data structures for range median queries. In: Proceedings of the International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 5878, pp. 822–831. Springer, Berlin (2009)

    Google Scholar 

  9. Brodal, G.S., Davoodi, P., Rao, S.S.: On space efficient two dimensional range minimum data structures. In: Proceedings of the European Symposium on Algorithms (ESA). Lecture Notes in Computer Science., vol. 6346/6347. Springer, Berlin (2010)

    Google Scholar 

  10. Brodal, G.S., Davoodi, P., Rao, S.S.: Path minima queries in dynamic weighted trees. In: Proceedings of the Workshop on Algorithms and Data Structures (WADS). Lecture Notes in Computer Science, vol. 6844, pp. 290–301. Springer, Berlin (2011)

    Chapter  Google Scholar 

  11. Brodal, G.S., Gfeller, B., Jørgensen, A.G., Sanders, P.: Towards optimal range medians. Theor. Comput. Sci. 412(24), 2588–2601 (2011)

    Article  MATH  Google Scholar 

  12. Brodal, G.S., Davoodi, P., Lewenstein, M., Raman, R., Rao, S.S.: Two dimensional range minimum queries and Fibonacci lattices. In: Proceedings of the European Symposium on Algorithms (ESA). Lecture Notes in Computer Science, vol. 7501, pp. 217–228. Springer, Berlin (2012)

    Google Scholar 

  13. Chan, T.M.: Optimal partition trees. Discrete Comput. Geom. 47(4), 661–690 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chan, T.M., Pătraşcu, M.: Counting inversions, offline orthogonal range counting, and related problems. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 161–173 (2010)

    Chapter  Google Scholar 

  15. Chan, T.M., Durocher, S., Larsen, K.G., Morrison, J., Wilkinson, B.T.: Linear-space data structures for range mode query in arrays. In: Proceedings of the International Symposium on Theoretical Aspects of Computer Science (STACS). Leibniz International Proceedings in Informatics, vol. 14, pp. 291–301 (2012)

    Google Scholar 

  16. Chan, T.M., Durocher, S., Skala, M., Wilkinson, B.T.: Linear-space data structures for range minority query in arrays. In: Proceedings of the Scandinavian Symposium and Workshops on Algorithm Theory (SWAT). Lecture Notes in Computer Science, vol. 7357, pp. 295–306. Springer, Berlin (2012)

    Google Scholar 

  17. Chazelle, B.: Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom. 9(2), 145–158 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chazelle, B.: Cuttings. In: Handbook of Data Structures and Applications, pp. 25.1–25.10. CRC Press, Boca Raton (2005)

    Google Scholar 

  19. Chazelle, B., Rosenberg, B.: Computing partial sums in multidimensional arrays. In: Proceedings of the ACM Symposium on Computational Geometry (SoCG), pp. 131–139 (1989)

    Google Scholar 

  20. Davoodi, P.: Data structures: range queries and space efficiency. Ph.D. thesis, Aarhus University (2011)

  21. Davoodi, P., Raman, R., Satti, S.R.: Succinct representations of binary trees for range minimum queries. In: Proceedings of the International Computing and Combinatorics Conference (COCOON). Lecture Notes in Computer Science, vol. 7434, pp. 396–407. Springer, Berlin (2012)

    Google Scholar 

  22. de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)

    Book  Google Scholar 

  23. Demaine, E.D., Landau, G.M., Weimann, O.: On Cartesian trees and range minimum queries. In: Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 5555, pp. 341–353. Springer, Berlin (2009)

    Chapter  Google Scholar 

  24. Dietz, P.F.: Optimal algorithms for list indexing and subset rank. In: Proceedings of the Workshop on Algorithms and Data Structures (WADS). Lecture Notes in Computer Science, vol. 382, pp. 39–46. Springer, Berlin (1989)

    Chapter  Google Scholar 

  25. Dobkin, D., Munro, J.I.: Determining the mode. Theor. Comput. Sci. 12(3), 255–263 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  26. Durocher, S.: A simple linear-space data structure for constant-time range minimum query. CoRR (2011). arXiv:1109.4460

  27. Durocher, S., He, M., Munro, J.I., Nicholson, P.K., Skala, M.: Range majority in constant time and linear space. In: Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 6755, pp. 244–255. Springer, Berlin (2011)

    Chapter  Google Scholar 

  28. Durocher, S., He, M., Munro, J.I., Nicholson, P.K., Skala, M.: Range majority in constant time and linear space. Inf. Comput. 222, 169–179 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Elmasry, A., He, M., Munro, J.I., Nicholson, P.: Dynamic range majority data structures. In: Proceedings of the International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 7074, pp. 150–159. Springer, Berlin (2011)

    Google Scholar 

  30. Fischer, J.: Optimal succinctness for range minimum queries. In: Proceedings of the Latin American Theoretical Informatics Symposium (LATIN). Lecture Notes in Computer Science, vol. 6034, pp. 158–169. Springer, Berlin (2010)

    Google Scholar 

  31. Fischer, J., Heun, V.: Theoretical and practical improvements on the RMQ-problem, with applications to LCA and LCE. In: Proceedings of the Symposium on Combinatorial Pattern Matching (CPM). Lecture Notes in Computer Science, vol. 4009, pp. 36–48. Springer, Berlin (2006)

    Chapter  Google Scholar 

  32. Fischer, J., Heun, V.: A new succinct representation of RMQ-information and improvements in the enhanced suffix array. In: Proceedings of the International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies (ESCAPE). Lecture Notes in Computer Science, vol. 4614, pp. 459–470. Springer, Berlin (2007)

    Chapter  Google Scholar 

  33. Fischer, J., Heun, V.: Finding range minima in the middle: approximations and applications. Math. Comput. Sci. 3(1), 17–30 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: Proceedings of the ACM Symposium on the Theory of Computing (STOC), pp. 135–143 (1984)

    Google Scholar 

  35. Gagie, T., Puglisi, S.J., Turpin, A.: Range quantile queries: another virtue of wavelet trees. In: Proceedings of the String Processing and Information Retrieval Symposium (SPIRE). Lecture Notes in Computer Science, vol. 5721, pp. 1–6. Springer, Berlin (2009)

    Chapter  Google Scholar 

  36. Gagie, T., He, M., Munro, J.I., Nicholson, P.: Finding frequent elements in compressed 2D arrays and strings. In: Proceedings of the Symposium on String Processing and Information Retrieval (SPIRE). Lecture Notes in Computer Science, vol. 7024, pp. 295–300. Springer, Berlin (2011)

    Chapter  Google Scholar 

  37. Gfeller, B., Sanders, P.: Towards optimal range medians. In: Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 5555, pp. 475–486. Springer, Berlin (2009)

    Chapter  Google Scholar 

  38. Golin, M.J., Iacono, J., Krizanc, D., Raman, R., Rao, S.S.: Encoding 2D range maximum queries. In: Proceedings of the International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 7074, pp. 180–189. Springer, Berlin (2011)

    Google Scholar 

  39. Greve, M., Jørgensen, A.G., Larsen, K.D., Truelsen, J.: Cell probe lower bounds and approximations for range mode. In: Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 6198, pp. 605–616. Springer, Berlin (2010)

    Chapter  Google Scholar 

  40. Har-Peled, S., Muthukrishnan, S.: Range medians. In: Proceedings of the European Symposium on Algorithms (ESA). Lecture Notes in Computer Science, vol. 5193, pp. 503–514. Springer, Berlin (2008)

    Google Scholar 

  41. He, M., Munro, J.I., Nicholson, P.: Dynamic range selection in linear space. In: Proceedings of the International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 7074, pp. 160–169. Springer, Berlin (2011)

    Google Scholar 

  42. JáJá, J., Mortensen, C.W., Shi, Q.: Space-efficient and fast algorithms for multidimensional dominance reporting and counting. In: Proceedings of the International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 3341, pp. 558–568. Springer, Berlin (2004)

    Google Scholar 

  43. Jørgensen, A.G.: Data structures: sequence problems, range queries, and fault tolerance. Ph.D. thesis, Aarhus University (2010)

  44. Jørgensen, A.G., Larsen, K.D.: Range selection and median: tight cell probe lower bounds and adaptive data structures. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 805–813 (2011)

    Chapter  Google Scholar 

  45. Krizanc, D., Morin, P., Smid, M.: Range mode and range median queries on lists and trees. In: Proceedings of the International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 2906, pp. 517–526. Springer, Berlin (2003)

    Google Scholar 

  46. Krizanc, D., Morin, P., Smid, M.: Range mode and range median queries on lists and trees. Nord. J. Comput. 12, 1–17 (2005)

    MathSciNet  MATH  Google Scholar 

  47. Larsen, K.G.: The cell probe complexity of dynamic range counting. In: Proceedings of the ACM Symposium on the Theory of Computing (STOC), pp. 85–94 (2012)

    Google Scholar 

  48. Matoušek, J.: Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10(2), 157–182 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  49. Munro, J.I.: Tables. In: Chandru, V., Vinay, V. (eds.) Foundations of Software Technology and Theoretical Computer Science. Lecture Notes in Computer Science, vol. 1180, pp. 37–42. Springer, Berlin (1996)

    Chapter  Google Scholar 

  50. Munro, J.I., Spira, M.: Sorting and searching in multisets. SIAM J. Comput. 5(1), 1–8 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  51. Nekrich, Y.: Orthogonal range searching in linear and almost-linear space. In: Proceedings of the Workshop on Algorithms and Data Structures (WADS). Lecture Notes in Computer Science, vol. 4619, pp. 15–26. Springer, Berlin (2007)

    Chapter  Google Scholar 

  52. Pǎtraşcu, M.: Towards polynomial lower bounds for dynamic problems. In: Proceedings of the ACM Symposium on the Theory of Computing (STOC), pp. 603–610 (2010)

    Google Scholar 

  53. Petersen, H.: Improved bounds for range mode and range median queries. In: Proceedings of the Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM). Lecture Notes in Computer Science, vol. 4910, pp. 418–423. Springer, Berlin (2008)

    Google Scholar 

  54. Petersen, H., Grabowski, S.: Range mode and range median queries in constant time and sub-quadratic space. Inf. Process. Lett. 109, 225–228 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  55. Poon, C.K.: Optimal range max datacub for fixed dimensions. In: Proceedings of the International Conference on Database Theory (ICDT). Lecture Notes in Computer Science, vol. 2572, pp. 158–172. Springer, Berlin (2003)

    Google Scholar 

  56. Sadakane, K.: Succinct data structures for flexible text retrieval systems. J. Discrete Algorithms 5, 12–22 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  57. Skiena, S.: The Algorithm Design Manual, 2nd edn. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  58. van Emde Boas, P.: Preserving order in a forest in less than logarithmic time and linear space. Inf. Process. Lett. 6(3), 80–82 (1977)

    Article  MATH  Google Scholar 

  59. Vassilevska Williams, V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of the ACM Symposium on the Theory of Computing (STOC), pp. 887–898 (2012)

    Google Scholar 

  60. Yao, A.C.: Space-time tradeoff for answering range queries. In: Proceedings of the ACM Symposium on the Theory of Computing (STOC), pp. 128–136 (1982)

    Google Scholar 

  61. Yao, A.C.: On the complexity of maintaining partial sums. SIAM J. Comput. 14, 277–288 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  62. Yuan, H., Atallah, M.J.: Data structures for range minimum queries. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 150–160 (2010)

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors thank Peyman Afshani, Francisco Claude, Meng He, Ian Munro, Patrick Nicholson, Matthew Skala, and Norbert Zeh for discussing various topics related to range searching.

Author information

Authors and Affiliations

  1. University of Waterloo, Waterloo, Canada

    Timothy M. Chan

  2. University of Manitoba, Winnipeg, Canada

    Stephane Durocher & Jason Morrison

  3. Aarhus University, Aarhus, Denmark

    Kasper Green Larsen & Bryan T. Wilkinson

Authors
  1. Timothy M. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephane Durocher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kasper Green Larsen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jason Morrison
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bryan T. Wilkinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephane Durocher.

Additional information

A preliminary version of these results appeared at the 29th Symposium on Theoretical Aspects of Computer Science (STACS) [15]. Work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), and in part by MADALGO—Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chan, T.M., Durocher, S., Larsen, K.G. et al. Linear-Space Data Structures for Range Mode Query in Arrays. Theory Comput Syst 55, 719–741 (2014). https://doi.org/10.1007/s00224-013-9455-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-013-9455-2

Keywords

Search

Navigation