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I/O-Efficient Batched Range Counting and Its Applications to Proximity Problems

  • Tamás Lukovszki
  • Anil Maheshwari
  • Norbert Zeh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2245)

Abstract

We present an algorithm to answer a set Q of range counting queries over a point set P in d dimensions. The algorithm takes \( O\left( {sort(\left| P \right| + \left| Q \right|) + \tfrac{{(\left| P \right| + \left| Q \right|)\alpha (\left| P \right|)}} {{DB}}\log _{M/B}^{d - 1} \tfrac{{\left| P \right| + \left| Q \right|}} {B}} \right)^1 \) I/Os and uses linear space. For an important special case, the α(∣P∣) term in the I/Ocomplexity of the algorithm can be eliminated. We apply this algorithm to constructing t-spanners for point sets in ℝd and polygonal obstacles in the plane, and finding the K closest pairs of a point set in ℝd.

Keywords

Query Range External Memory Range Counting Sweep Line Span Ratio 
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. 1.
    P. K. Agarwal, J. Erickson. Geometric range searching and its relatives. Advances in Disc. and Comp. Geom., pp. 1–56. AMS, 1999.Google Scholar
  2. 2.
    L. Arge, O. Procopiuc, S. Ramaswamy, T. Suel, J. S. Vitter. Theory and practice of I/O-efficient algorithms for multidimensional batched searching problems. Proc. SODA, 1998.Google Scholar
  3. 3.
    L. Arge, V. Samoladas, J. S. Vitter. On two-dimensional indexability and optimal range search indexing. Proc. PODS’99, 1999.Google Scholar
  4. 4.
    L. Arge. The buffer tree: A new technique for optimal I/O-algorithms. Proc. WADS, pp. 334–345, 1995.Google Scholar
  5. 5.
    L. Arge, P. B. Miltersen. On showing lower bounds for external-memory computational geometry problems. In J. Abello and J. S. Vitter (eds.), External Memory Algorithms and Visualization. AMS, 1999.Google Scholar
  6. 6.
    L. Arge, D. E. Vengro., J. S. Vitter. External-memory algorithms for processing line segments in geographic information systems. Proc. ESA, pp. 295–310, 1995.Google Scholar
  7. 7.
    L. Arge, J. S. Vitter. Optimal dynamic interval management in external memory. Proc. FOCS, pp. 560–569, 1996.Google Scholar
  8. 8.
    S. Arya, G. Das, D. M. Mount, J. S. Salowe, M. Smid. Euclidean spanners: Short, thin, and lanky. Proc. STOC, pp. 489–498, 1995.Google Scholar
  9. 9.
    Y.-J. Chiang, M. T. Goodrich, E. F. Grove, R. Tamassia, D. E. Vengro., J. S. Vitter. External-memory graph algorithms. Proc. SODA, 1995.Google Scholar
  10. 10.
    P. B. Callahan, S. R. Kosaraju. A decomposition of multi-dimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Proc. STOC, pp. 546–556, 1992.Google Scholar
  11. 11.
    K. L. Clarkson. Approximation algorithms for shortest path motion planning. Proc. STOC, pp. 56–65, 1987.Google Scholar
  12. 12.
    M. Fischer, T. Lukovszki, M. Ziegler. Geometric searching in walkthrough animations with weak spanners in real time. Proc. ESA, pp. 163–174, 1998.Google Scholar
  13. 13.
    M. T. Goodrich, J.-J. Tsay, D. E. Vengro., J. S. Vitter. External-memory computational geometry. Proc. FOCS, 1993.Google Scholar
  14. 14.
    S. Govindarajan, T. Lukovszki, A. Maheshwari, N. Zeh. I/O-efficient well-separated pair decomposition and its applications. Proc. ESA, pp. 220–231, 2000.Google Scholar
  15. 15.
    Y. Hassin, D. Peleg. Sparse communication networks and efficient routing in the plane. Proc. PODC, 2000.Google Scholar
  16. 16.
    J. M. Keil, C. A. Gutwin. Classes of graphs which approximate the complete Euclidean graph. Discrete & Computational Geometry, 7:13–28, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    C. Levcopoulos, G. Narasimhan, M. Smid. Efficient algorithms for constructing fault-tolerant geometric spanners. Proc. STOC, pp. 186–195, 1998.Google Scholar
  18. 18.
    T. Lukovszki. New results on fault tolerant geometric spanners. Proc. WADS, pp. 193–204, 1999.Google Scholar
  19. 19.
    J. Ruppert, R. Seidel. Approximating the d-dimensional complete Euclidean graph. Proc. CCCG, pp. 207–210, 1991.Google Scholar
  20. 20.
    D. E. Vengro., J. S. Vitter. Efficient 3-D range searching in external memory. Proc. STOC, 1996.Google Scholar
  21. 21.
    J. S. Vitter. External memory algorithms. Proc. PODS, pp. 119–128, 1998.Google Scholar
  22. 22.
    J. S. Vitter, E. A. M. Shriver. Algorithms for parallel memory I: Two-level memories. Algorithmica, 12(2—3):110–147, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. C. Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comp., 11:721–736, 1982.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Tamás Lukovszki
    • 1
  • Anil Maheshwari
    • 2
  • Norbert Zeh
    • 2
  1. 1.Heinz-Nixdorf-InstitutUniversität PaderbornGermany
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

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