Skip to main content
SpringerLink
Log in

Colored Range Searching in Linear Space

  • Conference paper
Algorithm Theory – SWAT 2014 (SWAT 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8503))

Included in the following conference series:

Abstract

In colored range searching, we are given a set of n colored points in d ≥ 2 dimensions to store, and want to support orthogonal range queries taking colors into account. In the colored range counting problem, a query must report the number of distinct colors found in the query range, while an answer to the colored range reporting problem must report the distinct colors in the query range.

We give the first linear space data structure for both problems in two dimensions (d = 2) with o(n) worst case query time. We also give the first data structure obtaining almost-linear space usage and o(n) worst case query time for points in d > 2 dimensions. Finally, we present the first dynamic solution to both counting and reporting with o(n) query time for d ≥ 2 and d ≥ 3 dimensions, respectively.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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.

Similar content being viewed by others

References

  1. Agarwal, P.K., Govindarajan, S., Muthukrishnan, S.M.: Range searching in categorical data: Colored range searching on grid. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 17–28. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  2. Andersson, A.: General balanced trees. J. Algorithms 30(1), 1–18 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Chapter  Google Scholar 

  5. van Emde Boas, P., Kaas, R., Zijlstra, E.: Design and implementation of an efficient priority queue. Theory Comput. Syst. 10(1), 99–127 (1976)

    Google Scholar 

  6. Gagie, T., Kärkkäinen, J., Navarro, G., Puglisi, S.J.: Colored range queries and document retrieval. TCS (2012)

    Google Scholar 

  7. 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  MATH  MathSciNet  Google Scholar 

  8. Gupta, P., Janardan, R., Smid, M.: A technique for adding range restrictions to generalized searching problems. Inform. Process. Lett. 64(5), 263–269 (1997)

    Article  MathSciNet  Google Scholar 

  9. JáJá, J., Mortensen, C.W., Shi, Q.: Space-efficient and fast algorithms for multidimensional dominance reporting and counting. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 558–568. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  10. Janardan, R., Lopez, M.: Generalized intersection searching problems. IJCGA 3(01), 39–69 (1993)

    MATH  MathSciNet  Google Scholar 

  11. Kaplan, H., Rubin, N., Sharir, M., Verbin, E.: Counting colors in boxes. In: Proc. 18th SODA. pp. 785–794 (2007)

    Google Scholar 

  12. van Kreveld, M.: New results on data structures in computational geometry. PhD thesis, Department of Computer Science, University of Utrecht, Netherlands (1992)

    Google Scholar 

  13. Larsen, K.G., Pagh, R.: I/O-efficient data structures for colored range and prefix reporting. In: Proc. 23rd SODA. pp. 583–592 (2012)

    Google Scholar 

  14. Larsen, K.G., van Walderveen, F.: Near-Optimal Range Reporting Structures for Categorical Data. In: Proc. 24th SODA. pp. 265–276 (2013)

    Google Scholar 

  15. Mehlhorn, K., Näher, S.: Bounded ordered dictionaries in O(loglogN) time and O(n) space. Inform. Process. Lett. 35(4), 183–189 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mortensen, C.W.: Generalized static orthogonal range searching in less space. Tech. rep., TR-2003-22, The IT University of Copenhagen (2003)

    Google Scholar 

  17. Nekrich, Y.: Orthogonal Range Searching in Linear and Almost-linear Space. Comput. Geom. Theory Appl. 42(4), 342–351 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nekrich, Y.: Space-efficient range reporting for categorical data. In: Proc. 31st PODS. pp. 113–120 (2012)

    Google Scholar 

  19. Nekrich, Y.: Efficient range searching for categorical and plain data. ACM TODS 39(1), 9 (2014)

    Article  MathSciNet  Google Scholar 

  20. Nekrich, Y., Vitter, J.S.: Optimal color range reporting in one dimension. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 743–754. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  21. Overmars, M.H.: Design of Dynamic Data Structures (1987)

    Google Scholar 

  22. Patrascu, M.: Lower bounds for 2-dimensional range counting. In: Proc. 39th STOC, pp. 40–46 (2007)

    Google Scholar 

  23. Shi, Q., JáJá, J.: Optimal and near-optimal algorithms for generalized intersection reporting on pointer machines. Inform. Process. Lett. 95(3), 382–388 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Willard, D.: Log-logarithmic worst-case range queries are possible in space Θ(N). Inform. Process. Lett. 17(2), 81–84 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  25. Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proc. 44th STOC. pp. 887–898 (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università di Pisa, Italy

    Roberto Grossi

  2. Technical University of Denmark, DTU Compute, Denmark

    Søren Vind

Authors
  1. Roberto Grossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Søren Vind
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Tepper School of Business, Carnegie Mellon University, 15213, Pittsburgh, PA, USA

    R. Ravi

  2. DTU Informatics, 2800, Kongens Lyngby, Denmark

    Inge Li Gørtz

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Grossi, R., Vind, S. (2014). Colored Range Searching in Linear Space. In: Ravi, R., Gørtz, I.L. (eds) Algorithm Theory – SWAT 2014. SWAT 2014. Lecture Notes in Computer Science, vol 8503. Springer, Cham. https://doi.org/10.1007/978-3-319-08404-6_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-08404-6_20

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08403-9

  • Online ISBN: 978-3-319-08404-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation