# Shallow interdistance selection and interdistance enumeration

## Abstract

Shallow interdistance selection refers to the problem of selecting the *k*^{ th } smallest interdistance, *k*≤*n*, 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 2*x*. 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## Preview

Unable to display preview. Download preview PDF.

## 6. References

- 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.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.B. Chazelle, Some Techniques for Geometric Searching with Implicit Set Representations,
*Acta Informatica,***24**, 1987, pp. 565–582.Google Scholar - 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.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.F. P. Preparata and M. I. Shamos,
*Computational Geometry: An Introduction*, Springer Verlag, New York, NY, 1985.Google Scholar - 7.J. S. Salowe, L-Infinity Interdistance Selection by Parametric Search,
*Inf. Proc. Letters,***30**, 1989, pp. 9–14.Google Scholar - 8.M. Smid, Maintaining the Minimal Distance of a Point Set in Polylogarithmic Time,
*Universitat des Saarlandes 13/90*, 1990.Google Scholar - 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.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.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