## Fast Algorithms for Approximating Distances

Abstract: Let S be a set of n points in the plane and let 1 \leq k \leq n(n-1)/2. Let d_1 \leq d_2 \leq ... \leq d (n choose 2) be the Lp-distances determined by the pairs of points in S. In this paper we consider the following optimization problems:

• Distance selection. Compute the k-th smallest distance between a pair of points of S.
• Distance ranking. Determine how many pair of points are closer than unit distance apart.
• Distance by query. Preprocess S into a data structure, so that, given a number k as above, one can answer efficiently what is the k-th smallest distance between a pair of points of S.
We present algorithms for approximate distance selection with running time O(n lg3n) (in contrast with the best known O(n4/3polylogn) randomized algorithm for the exact problem). We also provide efficient approximation algorithms for distance ranking and distance queries.