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Shallow interdistance selection and interdistance enumeration

  • Jeffrey S. Salowe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 519)

Abstract

Shallow interdistance selection refers to the problem of selecting the k th smallest interdistance, kn, from among the \(\left( {\begin{array}{*{20}c}n \\2 \\\end{array} } \right)\) interdistances determined by a set of n points in \(\Re ^d\). Shallow interdistance selection has a concrete application — it is a crucial component in the design of a linear-sized data structure that dynamically maintains the minimum interdistance in sublinear time per operation (Smid [9]). In addition, the study of shallow interdistance selection may provide insight into developing more efficient algorithms for the problem of selecting Euclidean interdistances (Agarwal et al. [1]). We give a shallow interdistance selection algorithm which takes optimal O(n log n) time and works in any L p metric. To do this, we prove two interesting related results. The first is a combinatorial result relating the rank of x to the rank of 2x. The second is an algorithm which enumerates all pairs of points within interdistance x in time proportional to the rank of x (plus O(n log n)). A corollary to our work is an algorithm which, given a set of n points and an integer k, outputs all interdistances having rank at most k in O(n log n+k) time.

Keywords

Selection Algorithm Minimum Span Tree Neighbor Algorithm Combinatorial Result Neighbor Problem 
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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6. References

  1. 1.
    P. K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting Distances in the Plane, Sixth ACM Symposium on Computational Geometry, 1990, pp. 321–331.Google Scholar
  2. 2.
    J. Bentley, D. Stanat and E. Williams, The Complexity of Finding Fixed-Radius Near Neighbors, Inf. Proc. Letters, 6, 1977, pp. 209–213.Google Scholar
  3. 3.
    B. Chazelle, Some Techniques for Geometric Searching with Implicit Set Representations, Acta Informatica, 24, 1987, pp. 565–582.Google Scholar
  4. 4.
    M. T. Dickerson and R. L. S. Drysdale, Fixed-Radius Near Neighbors Search Algorithms for Points and Segments, Inf. Proc. Letters, 35, 1990, pp. 269–273.Google Scholar
  5. 5.
    M. T. Dickerson and R. L. S. Drysdale, Enumerating k Distances for n Points in the Plane, Seventh ACM Symposium on Computational Geometry, 1991.Google Scholar
  6. 6.
    F. P. Preparata and M. I. Shamos, Computational Geometry: An Introduction, Springer Verlag, New York, NY, 1985.Google Scholar
  7. 7.
    J. S. Salowe, L-Infinity Interdistance Selection by Parametric Search, Inf. Proc. Letters, 30, 1989, pp. 9–14.Google Scholar
  8. 8.
    M. Smid, Maintaining the Minimal Distance of a Point Set in Polylogarithmic Time, Universitat des Saarlandes 13/90, 1990.Google Scholar
  9. 9.
    M. Smid, Maintaining the Minimal Distance of a Point Set in Less Than Linear Time, Universitat des Saarlandes 06/90, 1990.Google Scholar
  10. 10.
    P. M. Vaidya, An O(n log n) Algorithm for the All-Nearest-Neighbors Problem, Discrete Comput. Geom., 4, 1989, pp. 101–115.Google Scholar
  11. 11.
    A. C. Yao, On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems, Siam J. on Computing, 11, 1982, pp. 721–736.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Jeffrey S. Salowe
    • 1
  1. 1.Department of Computer ScienceUniversity of VirginiaCharlottesville

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