GeoInformatica

, Volume 18, Issue 2, pp 193–228 | Cite as

The largest empty rectangle containing only a query object in Spatial Databases

  • Gilberto Gutiérrez
  • José R. Paramá
  • Nieves Brisaboa
  • Antonio Corral
Article

Abstract

Let S be a set of n points in a fixed axis-parallel rectangle \(R\subseteq \Re^{2}\), i.e. in the two-dimensional space (2D). Assuming that those points are stored in an R-tree, this paper presents several algorithms for finding the empty rectangle in R with the largest area, sides parallel to the axes of the space, and containing only a query point q. This point can not be part of S, that is, it is not stored in the R-tree. All algorithms follow the basic idea of discarding part of the points of S, in such a way that the problem can be solved only considering the remaining points. As a consequence, the algorithms only have to access a very small portion of the nodes (disk blocks) of the R-tree, saving main memory resources and computation time. We provide formal proofs of the correctness of our algorithms and, in order to evaluate the performance of the algorithms, we run an extensive set of experiments using synthetic and real data. The results have demonstrated the efficiency and scalability of our algorithms for different dataset configurations.

Keywords

Spatial databases Query Indexing methods 

References

  1. 1.
    Aggarwal A, Suri S (1987) Fast algorithms for computing the largest empty rectangle. In: Proceedings of SCG ’87. ACM, pp 278–290Google Scholar
  2. 2.
    Augustine J, Das S, Maheshwari A, Nandy SC, Roy S, Sarvattomananda S (2010) Recognizing the largest empty circle and axis-parallel rectangle in a desired location. CoRR abs/1004.0558
  3. 3.
    Augustine J, Das S, Maheshwari A, Nandy SC, Roy S, Sarvattomananda S (2010) Querying for the largest empty geometric object in a desired location. CoRR abs/1004.0558v2
  4. 4.
    Augustine J, Putnam B, Roy S (2010) Largest empty circle centered on a query line. J Discrete Algorithms 8(2):143–153CrossRefGoogle Scholar
  5. 5.
    Augustine J, Das S, Maheshwari A, Nandy SC, Roy S, Sarvattomananda S (2013) Localized geometric query problems. Comput Geom 46(3):340–357CrossRefGoogle Scholar
  6. 6.
    Beckmann N, Kriegel H, Schneider R, Seeger B (1990) The R*-tree: An efficient and robust access method for points and rectangles, In: Garcia-Molina H, Jagadish HV (eds) Proceedings of SIGMOD ’09. ACM Press, pp 322–331Google Scholar
  7. 7.
    Böhm C, Kriegel H.-P (2001) Determining the convex hull in large multidimensional databases. In: Proceedings of DaWaK ’01. Springer, pp 294–306Google Scholar
  8. 8.
    Börzsönyi S, Kossmann D, Stocker K (2001) The skyline operator. In: Proccedings of ICDE ’01. pp 421–430Google Scholar
  9. 9.
    Chazelle B, Drysdalet RL, Lee DT (1986) Computing the largest empty rectangle. SIAM J Comput 15:300–315CrossRefGoogle Scholar
  10. 10.
    Chew LP, Drysdale RLS (1986) Finding largest empty circles with location constraints. Tech. Rep. PCS-TR86-130, Dartmouth College, Computer Science, Hanover, NHGoogle Scholar
  11. 11.
    Corral A, Manolopoulos Y, Theodoridis Y, Vassilakopoulos M (2004) Algorithms for processing k-closest-pair queries in spatial databases. Data Knowl Eng 49(1):67–104CrossRefGoogle Scholar
  12. 12.
    De M, Nandy SC (2011) Inplace algorithm for priority search tree and its use in computing largest empty axis-parallel rectangle. CoRR abs/1104.3076
  13. 13.
    Dellis E, Seeger B (2007) Efficient computation of reverse skyline queries. In: Proceedings of VLDB ’07. ACM, pp 291–302Google Scholar
  14. 14.
    Edmonds J, Gryz J, Liang D, Miller RJ (2003) Mining for empty spaces in large data sets. Theor Comput Sci 296:435–452CrossRefGoogle Scholar
  15. 15.
    Gaede V, Günther O (1998) Multidimensional access methods. ACM Comput Surv 30(2):170–231CrossRefGoogle Scholar
  16. 16.
    Gutiérrez G, Paramá J (2012) Finding the largest empty rectangle containing only a query point in large multidimensional databases. In: Proceedings of SSDBM 2012. SpringerGoogle Scholar
  17. 17.
    Guttman A (1984) R-trees: A dynamic index structure for spatial searching. In: Proceedings of SIGMOD ’84, ACM, pp 47–57Google Scholar
  18. 18.
    Hjaltason GR, Samet H (1998) Incremental distance join algorithms for spatial databases. In: Proceedings of SIGMOD ’98, ACM, pp 237–248Google Scholar
  19. 19.
    Kaplan H, Mozes S, Nussbaum Y, Sharir M (2012) Submatrix maximum queries in monge matrices and monge partial matrices, and their applications. In: Proceedings of SODA 2012, SIAM, pp 338–355Google Scholar
  20. 20.
    Kaplan H, Sharir M (2012) Finding the maximal empty disk containing a query point. In: Proceedings of SCG 2012, SoCG ’12. ACM, New York, NY, USA, pp 287–292Google Scholar
  21. 21.
    Manolopoulos Y, Nanopoulos A, Papadopoulos AN, Theodoridis Y (2005) R-Trees: Theory and Applications (Advanced information and knowledge processing). Springer-Verlag New York, Inc., Secaucus, NJ, USAGoogle Scholar
  22. 22.
    Minati D, Nandy S (2011) Space-efficient algorithms for empty space recognition among a point set in 2d and 3d. In: Proceedings of the 23rd annual Canadian conference on computational geometry, pp 347–353Google Scholar
  23. 23.
    Naamad A, Lee DT, Hsu W-L (1984) On the maximum empty rectangle problem. Discrete Appl Math 8:267–277CrossRefGoogle Scholar
  24. 24.
    Nandy S, Bhattacharya B (1998) Maximal empty cuboids among points and blocks. Comput Math Appl 36(3):11–20CrossRefGoogle Scholar
  25. 25.
    Oracle spatial user’s guide and reference (2012) http://docs.oracle.com/html/A88805_01/sdo_intr.htm
  26. 26.
    Orlowski M (1990) A new algorithm for the largest empty rectangle problem. Algorithmica 5:65–73CrossRefGoogle Scholar
  27. 27.
    Papadias D, Tao Y, Fu G, Seeger B (2005) Progressive skyline computation in database systems. ACM T Database Syst 30(1):41–82CrossRefGoogle Scholar
  28. 28.
  29. 29.
    Roussopoulos N, Kelley S, Vincent F (1995) Nearest neighbor queries. SIGMOD Rec 24(2):71–79CrossRefGoogle Scholar
  30. 30.
    Shekhar S, Chawla S (2003) Spatial databases - a tour. Prentice HallGoogle Scholar
  31. 31.
    Toussaint GT (1983) Computing largest empty circles with location constraints. In J Comput Inf Sci 12(5):347–358CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Gilberto Gutiérrez
    • 1
  • José R. Paramá
    • 2
  • Nieves Brisaboa
    • 2
  • Antonio Corral
    • 3
  1. 1.Computer Science and Information Technologies DepartmentUniversidad del Bío-BíoChillánChile
  2. 2.Computer Science DepartmentUniversity of A CoruñaA CoruñaSpain
  3. 3.Department of Languages and ComputationUniversity of AlmeriaAlmeriaSpain

Personalised recommendations