Computing the Closest Point to a Circle

In this paper we consider the problem of computing the closest point to the boundary of a circle among a set S of n points. We present two algorithms to solve this problem. One algorithm runs in O(n) preprocessing time and space and O(log n) query time. The other algorithm runs in O(n ) preprocessing time and O(n log n) space and O(n ) query time. Thus we exhibit a trade-off between preprocessing and query times For dimensions d ≥ 3 we present an algorithm with O(n d/2 + ′ ) preprocessing time to report an approximate closest point to the boundary of d-dimensional query sphere R in O(n1−1/(d+1)+ ) query time.