ENUMERATING INTERDISTANCES IN SPACE

Abstract

In this paper, we study the enumerative versions of the interdistance in terdis stance ranking and selection problems in space, namely, the fixed-radius near neighbors and interdistance enumeration problems, respectively. The input to the fixed-radius near neighbors problem is a set of n points S⊆ℜd and a nonnegative real number δ, and the output consists of all pairs of points within interdistance δ. We give an algorithm which, after an O(n log n) time preprocessing step, answers a fixed-radius near neighbors query with respect to an Lp metric in O(n+ρ(δ)) time, where ρ(δ) is the rank of δ. The space needed is O(n). The input to the interdistance enumeration problem is a set of n points S⃄ℜd and an integer k, [Formula: see text], and the output is a set of point pairs, each corresponding to an interdistance having length less than or equal to the interdistance with rank k. We offer an O(n log n+k) time, O(n+k) space algorithm for this problem. This algorithm also works for any Lp metric.