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 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.