Skip to main content

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8066))

Abstract

Array range queries are of current interest in the field of data structures. Given an array of numbers or arbitrary elements, the general array range query problem is to build a data structure that can efficiently answer queries of a given type stated in terms of an interval of the indices. The specific query type might be for the minimum element in the range, the most frequently occurring element, or any of many other possibilities. In addition to being interesting in themselves, array range queries have connections to computational geometry, compressed and succinct data structures, and other areas of computer science. We survey recent and relevant past work on this class of problems.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agarwal, P.K., Erickson, J.: Geometric range searching and its relatives. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry. Contemporary Mathematics, vol. 223, pp. 1–56. American Mathematical Society, Providence (1999)

    Chapter  Google Scholar 

  2. Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.): ICALP 2009, Part I. LNCS, vol. 5555. Springer, Heidelberg (2009)

    Google Scholar 

  3. Alstrup, S., Brodal, G.S., Rauhe, T.: New data structures for orthogonal range searching. In: 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, Redondo Beach, California, USA, November 12-14, pp. 198–207 (2000)

    Google Scholar 

  4. Amir, A., Fischer, J., Lewenstein, M.: Two-dimensional range minimum queries. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 286–294. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  5. Asano, T., Nakano, S.-I., Okamoto, Y., Watanabe, O. (eds.): ISAAC 2011. LNCS, vol. 7074. Springer, Heidelberg (2011)

    Google Scholar 

  6. 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 

  7. Ben-Or, M.: Lower bounds for algebraic computation trees (preliminary report). In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, Boston, Massachusetts, April 25-27, pp. 80–86 (1983)

    Google Scholar 

  8. 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 

  9. Berkman, O., Vishkin, U.: Recursive *-tree parallel data-structure. In: Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, FOCS 1989, Research Triangle Park, NC, October 30-November 1, pp. 196–202. IEEE Computer Society Press, Los Alamitos (1989)

    Google Scholar 

  10. Berkman, O., Vishkin, U.: Recursive star-tree parallel data structure. SIAM J. Comput. 22(2), 221–242 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Chapter  Google Scholar 

  12. Bose, P., Kranakis, E., Morin, P., Tang, Y.: Approximate range mode and range median queries. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 377–388. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  13. Bozanis, P., Kitsios, N., Makris, C., Tsakalidis, A.: New upper bounds for generalized intersection searching problems. In: Fülöp, Z., Gécseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 464–474. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  14. Brodal, G.S., Davoodi, P., Rao, S.S.: On space efficient two dimensional range minimum data structures. Algorithmica 63(4), 815–830 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brodal, G.S., Fagerberg, R., Greve, M., López-Ortiz, A.: Online sorted range reporting. In: [28], pp. 173–182

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Brodal, G.S., Jørgensen, A.G.: Data structures for range median queries. In: [28], pp. 822–831

    Google Scholar 

  18. Brodnik, A., Karlsson, J., Munro, J.I., Nilsson, A.: An O(1) solution to the prefix sum problem on a specialized memory architecture. In: Navarro, G., Bertossi, L.E., Kohayakawa, Y. (eds.) TCS 2006. IFIP, vol. 209, pp. 103–114. Springer, Boston (2006)

    Google Scholar 

  19. Chan, T.M., Durocher, S., Larsen, K.G., Morrison, J., Wilkinson, B.T.: Linear-space data structures for range mode query in arrays. In: Dürr, C., Wilke, T. (eds.) 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, Paris, France, February 29-March 3. LIPIcs, vol. 14, pp. 290–301. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)

    Google Scholar 

  20. Chan, T.M., Durocher, S., Larsen, K.G., Morrison, J., Wilkinson, B.T.: Linear-space data structures for range mode query in arrays. Theory Comput. Sys., 1–23 (2013)

    Google Scholar 

  21. Chan, T.M., Durocher, S., Skala, M., Wilkinson, B.T.: Linear-space data structures for range minority query in arrays. In: [36], pp. 295–306

    Google Scholar 

  22. Chazelle, B.: A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput. 17(3), 427–462 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chazelle, B., Rosenberg, B.: The complexity of computing partial sums off-line. Internat. J. Comput. Geom. Appl. 1(1), 33–45 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chun, S.J., Chung, C.W., Lee, J.H., Lee, S.L.: Dynamic update cube for range-sum queries. In: Apers, P.M.G., Atzeni, P., Ceri, S., Paraboschi, S., Ramamohanarao, K., Snodgrass, R.T. (eds.) Proceedings of the Twenty-seventh International Conference on Very Large Data Bases, Roma, Italy, September 11-14, pp. 521–530. Morgan Kaufmann Publishers (2001)

    Google Scholar 

  25. de Berg, M., Haverkort, H.J.: Significant-presence range queries in categorical data. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 462–473. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  26. Demaine, E.D., Landau, G.M., Weimann, O.: On Cartesian trees and range minimum queries. In: [2], pp. 341–353

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Dong, Y., Du, D.-Z., Ibarra, O. (eds.): ISAAC 2009. LNCS, vol. 5878. Springer, Heidelberg (2009)

    Google Scholar 

  29. Durocher, S.: A simple linear-space data structure for constant-time range minimum query. In: Brodnik, A., López-Ortiz, A., Raman, V., Viola, A. (eds.) Munro Festschrift 2013. LNCS, vol. 8066, pp. 48–60. Springer, Heidelberg (2013)

    Google Scholar 

  30. 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)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Elmasry, A., He, M., Munro, J.I., Nicholson, P.K.: Dynamic range majority data structures. In: [5], pp. 150–159

    Google Scholar 

  33. Eppstein, D. (ed.): Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Mathematics (SODA 2002), January 6-8. ACM Press, New York (2002)

    Google Scholar 

  34. Fischer, J.: Optimal succinctness for range minimum queries. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 158–169. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  35. Fischer, J., Heun, V.: A new succinct representation of RMQ-information and improvements in the enhanced suffix array. In: Chen, B., Paterson, M., Zhang, G. (eds.) ESCAPE 2007. LNCS, vol. 4614, pp. 459–470. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  36. Fomin, F.V., Kaski, P. (eds.): SWAT 2012. LNCS, vol. 7357. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  37. Fredman, M.L.: The complexity of maintaining an array and computing its partial sums. J. ACM 29(1), 250–260 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, April 30-May 2, pp. 135–143. ACM Press, Washington, DC (1984)

    Google Scholar 

  39. 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)

    Chapter  Google Scholar 

  40. Gagie, T., Kärkkäinen, J.: Counting colours in compressed strings. In: Giancarlo, R., Manzini, G. (eds.) CPM 2011. LNCS, vol. 6661, pp. 197–207. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  41. Gagie, T., Kärkkäinen, J., Navarro, G., Puglisi, S.J.: Colored range queries and document retrieval. Theoret. Comput. Sci. 483, 36–50 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gagie, T., Navarro, G., Puglisi, S.J.: Colored range queries and document retrieval. In: Chavez, E., Lonardi, S. (eds.) SPIRE 2010. LNCS, vol. 6393, pp. 67–81. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  43. Gagie, T., Puglisi, S.J., Turpin, A.: Range quantile queries: Another virtue of wavelet trees. In: Karlgren, J., Tarhio, J., Hyyrö, H. (eds.) SPIRE 2009. LNCS, vol. 5721, pp. 1–6. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  44. Geffner, S., Agrawal, D., Abbadi, A.E., Smith, T.R.: Relative prefix sums: An efficient approach for querying dynamic OLAP data cubes. In: Kitsuregawa, M., Papazoglou, M.P., Pu, C. (eds.) Proceedings of the 15th International Conference on Data Engineering, Sydney, Austrialia, March 23-26, pp. 328–335. IEEE Computer Society (1999)

    Google Scholar 

  45. Gfeller, B., Sanders, P.: Towards optimal range medians. In: [2], pp. 475–486

    Google Scholar 

  46. Golynski, A.: Optimal lower bounds for rank and select indexes. Theoret. Comput. Sci. 387(3), 348–359 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  47. Greve, M., Jørgensen, A.G., Larsen, K.D., Truelsen, J.: Cell probe lower bounds and approximations for range mode. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 605–616. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  48. Grossi, R., Gupta, A., Vitter, J.S.: High-order entropy-compressed text indexes. In: SODA: ACM-SIAM Symposium on Discrete Algorithms (A Conference on Theoretical and Experimental Analysis of Discrete Algorithms), pp. 841–850. ACM/SIAM (2003)

    Google Scholar 

  49. Gupta, P., Janardan, R., Smid, M.: Further results on generalized intersection searching problems: Counting, reporting, and dynamization. J. Algorithms 19(2), 282–317 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  50. Hampapuram, H., Fredman, M.L.: Optimal biweighted binary trees and the complexity of maintaining partial sums. SIAM J. Comput. 28(1), 1–9 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  51. Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13(2), 338–355 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  52. He, M., Munro, J.I.: Space efficient data structures for dynamic orthogonal range counting. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 500–511. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  53. He, M., Munro, J.I., Nicholson, P.K.: Dynamic range selection in linear space. In: [5], pp. 160–169

    Google Scholar 

  54. He, M., Munro, J.I., Nicholson, P.K.: Dynamic range selection in linear space. CoRR abs/1106.5076v3 (2013), http://arxiv.org/abs/1106.5076v3

  55. He, M., Munro, J.I., Zhou, G.: Path queries in weighted trees. In: [5], pp. 140–149

    Google Scholar 

  56. He, M., Munro, J.I., Zhou, G.: Succinct data structures for path queries. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 575–586. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  57. Ho, C.T., Agrawal, R., Megiddo, N., Srikant, R.: Range queries in OLAP data cubes. In: Peckman, J.M. (ed.) Proceedings, ACM SIGMOD International Conference on Management of Data: SIGMOD 1997, Tucson, Arizona, USA, May 13-15. SIGMOD Record (ACM Special Interest Group on Management of Data), vol. 26(2), pp. 73–88. ACM Press (1997)

    Google Scholar 

  58. Hon, W.K., Shah, R., Thankachan, S.V.: Towards an optimal space-and-query-time index for top-k document retrieval. In: [64], pp. 173–184

    Google Scholar 

  59. Hon, W.K., Shah, R., Vitter, J.S.: Space-efficient framework for top-k string retrieval problems. In: 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, Atlanta, Georgia, USA, October 25-27, pp. 713–722. IEEE Computer Society (2009)

    Google Scholar 

  60. Jacobson, G.: Space-efficient static trees and graphs. In: 30th Annual Symp. on Foundations of Computer Science, vol. 30, pp. 549–554 (1989)

    Google Scholar 

  61. Jacobson, G.: Succinct Static Data Structures. PhD thesis, Carnegie-Mellon, Technical Report CMU-CS-89-112 (January 1989)

    Google Scholar 

  62. Janardan, R., Lopez, M.: Generalized intersection searching problems. Internat. J. Comput. Geom. Appl. 3(1), 39–69 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  63. Jørgensen, A.G., Larsen, K.G.: Range selection and median: Tight cell probe lower bounds and adaptive data structures. In: [83], pp. 805–813

    Google Scholar 

  64. Kärkkäinen, J., Stoye, J. (eds.): CPM 2012. LNCS, vol. 7354. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  65. Karpinski, M., Nekrich, Y.: Searching for frequent colors in rectangles. In: Proceedings of the 20th Annual Canadian Conference on Computational Geometry, Montréal, Canada, August 13-15 (2008)

    Google Scholar 

  66. Karpinski, M., Nekrich, Y.: Top-k color queries for document retrieval. In: [83], pp. 401–411

    Google Scholar 

  67. 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)

    Chapter  Google Scholar 

  68. 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)

    MathSciNet  MATH  Google Scholar 

  69. Lai, Y., Poon, C., Shi, B.: Approximate colored range and point enclosure queries. J. Discrete Algorithms 6(3), 420–432 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  70. Mäkinen, V., Navarro, G.: Rank and select revisited and extended. Theoret. Comput. Sci. 387(3), 332–347 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  72. Munro, J.I.: Tables. In: Chandru, V., Vinay, V. (eds.) FSTTCS 1996. LNCS, vol. 1180, pp. 37–42. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  73. Muthukrishnan, S.: Efficient algorithms for document retrieval problems. In: [33], pp. 657–666

    Google Scholar 

  74. Navarro, G.: Wavelet trees for all. In: [64], pp. 2–26

    Google Scholar 

  75. Nekrich, Y.: Orthogonal range searching in linear and almost-linear space. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 15–26. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  76. Nekrich, Y.: Orthogonal range searching in linear and almost-linear space. Comput. Geom. 42(4), 342–351 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  77. Nekrich, Y., Navarro, G.: Sorted range reporting. In: [36], pp. 271–282

    Google Scholar 

  78. Patil, M., Shah, R., Thankachan, S.V.: Succinct representations of weighted trees supporting path queries. J. Discrete Algorithms 17, 103–108 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  79. Patrascu, M.: Succincter. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, Philadelphia, Pennsylvania, USA, October 25-23, pp. 305–313. IEEE (2008)

    Google Scholar 

  80. Petersen, H.: Improved bounds for range mode and range median queries. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 418–423. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  81. Petersen, H., Grabowski, S.: Range mode and range median queries in constant time and sub-quadratic space. Inf. Process. Lett. 109(4), 225–228 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  82. Poon, C.K.: Dynamic orthogonal range queries in OLAP. Theoret. Comput. Sci. 296(3), 487–510 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  83. Randall, D. (ed.): Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25. SIAM (2011)

    Google Scholar 

  84. Sadakane, K.: Succinct representations of lcp information and improvements in the compressed suffix arrays. In: [35], pp. 225–232

    Google Scholar 

  85. Yao, A.C.C.: On the complexity of maintaining partial sums. SIAM J. Comput. 14(2), 277–288 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Skala, M. (2013). Array Range Queries. In: Brodnik, A., López-Ortiz, A., Raman, V., Viola, A. (eds) Space-Efficient Data Structures, Streams, and Algorithms. Lecture Notes in Computer Science, vol 8066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40273-9_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40273-9_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40272-2

  • Online ISBN: 978-3-642-40273-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics