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Finding the Largest Empty Rectangle Containing Only a Query Point in Large Multidimensional Databases

  • Gilberto Gutiérrez
  • José R. Paramá
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7338)

Abstract

Given a two-dimensional space, let S be a set of points stored in an R-tree, let R be the minimum rectangle containing the elements of S, and let q be a query point such that q ∉ S and R ∩ q ≠ ∅. In this paper, we present an algorithm for finding the empty rectangle with the largest area, sides parallel to the axes of the space, and containing only the query point q. The idea behind algorithm is to use the points that define the minimum bounding rectangles (MBRs) of some internal nodes of the R-tree to avoid reading as many nodes of the R-tree as possible, given that a naive algorithm must access all of them. We present several experiments considering synthetic and real data. The results show that our algorithm uses around 0.71–38% of the time and around 3–4% of the main storage needed by previous computational geometry algorithms. Furthermore, to the best of our knowledge, this is the first work that solves this problem considering that the points are stored in an R-tree.

Keywords

Geographical Information System Main Memory Query Point Real Point Naive Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [ADM+10a]
    Augustine, J., Das, S., Maheshwari, A., Nandy, S.C., Roy, S., Sarvattomananda, S.: Recognizing the largest empty circle and axis-parallel rectangle in a desired location. CoRR, abs/1004.0558 (2010)Google Scholar
  2. [ADM+10b]
    Augustine, J., Das, S., Maheshwari, A., Nandy, S.C., Roy, S., Sarvattomananda, S.: Querying for the largest empty geometric object in a desired location. CoRR, abs/1004.0558v2 (2010)Google Scholar
  3. [AS87]
    Aggarwal, A., Suri, S.: Fast algorithms for computing the largest empty rectangle. In: Proceedings of the Third Annual Symposium on Computational Geometry, SCG 1987, pp. 278–290. ACM, New York (1987)CrossRefGoogle Scholar
  4. [BK01]
    Böhm, C., Kriegel, H.-P.: Determining the Convex Hull in Large Multidimensional Databases. In: Kambayashi, Y., Winiwarter, W., Arikawa, M. (eds.) DaWaK 2001. LNCS, vol. 2114, pp. 294–306. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. [CDL86]
    Chazelle, B., Drysdalet, R.L., Lee, D.T.: Computing the largest empty rectangle. SIAM Journal Computing 15, 300–315 (1986)zbMATHCrossRefGoogle Scholar
  6. [CMTV04]
    Corral, A., Manolopoulos, Y., Theodoridis, Y., Vassilakopoulos, M.: Algorithms for processing k-closest-pair queries in spatial databases. Data & Knowledge Engineering 49(1), 67–104 (2004)CrossRefGoogle Scholar
  7. [Cor02]
    Corral, A.: Algoritmos para el Procesamiento de Consultas Espaciales utilizando R-trees. La Consulta de los Pares Más Cercanos y su Aplicación en Bases de Datos Espaciales. PhD thesis, Universidad de Almería, Escuela Politécnica Superior, España, Enero (2002)Google Scholar
  8. [DN11]
    De, M., Nandy, S.C.: Inplace algorithm for priority search tree and its use in computing largest empty axis-parallel rectangle. CoRR, abs/1104.3076 (2011)Google Scholar
  9. [EGLM03]
    Edmonds, J., Gryz, J., Liang, D., Miller, R.J.: Mining for empty spaces in large data sets. Theoretical Computer Science 296, 435–452 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [GG98]
    Gaede, V., Günther, O.: Multidimensional access methods. ACM Computing Surveys 30(2), 170–231 (1998)CrossRefGoogle Scholar
  11. [Gut84]
    Guttman, A.: R-trees: A dynamic index structure for spatial searching. In: ACM SIGMOD Conference on Management of Data, pp. 47–57. ACM (1984)Google Scholar
  12. [HS98]
    Hjaltason, G.R., Samet, H.: Incremental distance join algorithms for spatial databases. In: ACM SIGMOD Conference on Management of Data, Seattle, WA, pp. 237–248 (1998)Google Scholar
  13. [Kin81]
    King, J.J.: Query optimization by semantic reasoning. PhD thesis, Stanford University, CA, USA (1981)Google Scholar
  14. [KS11]
    Kaplan, H., Sharir, M.: Finding the maximal empty rectangle containing a query point. CoRR, abs/1106.3628 (2011)Google Scholar
  15. [NLH84]
    Naamad, A., Lee, D.T., Hsu, W.-L.: On the maximum empty rectangle problem. Discrete Applied Mathematics 8, 267–277 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [Orl90]
    Orlowski, M.: A new algorithm for the largest empty rectangle problem. Algorithmica 5, 65–73 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  17. [RKV95]
    Roussopoulos, N., Kelley, S., Vincent, F.: Nearest neighbor queries. In: SIGMOD 1995: Proceedings of the 1995 ACM SIGMOD International Conference on Management of Data, pp. 71–79. ACM Press, New York (1995)CrossRefGoogle Scholar
  18. [SC03]
    Shekhar, S., Chawla, S.: Spatial databases - a tour. Prentice Hall (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gilberto Gutiérrez
    • 1
  • José R. Paramá
    • 2
  1. 1.Departamento de Ciencias de la Computación y Tecnologías de la InformaciónUniversidad del Bío-BíoChillánChile
  2. 2.Department of Computer ScienceUniversity of A CoruñaEspaña

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