Closest pair and the post office problem for stochastic points ?
Given a (master) set M of n points in d -dimensional Euclidean space, consider drawing a random subset that includes each point mi?M m i ? M with an independent probability pi p i . How difficult is it to compute elementary statistics about the closest pair of points in such a subset? For instance, what is the probability that the distance between the closest pair of points in the random subset is no more than ?, for a given value ?? Or, can we preprocess the master set M such that given a query point q, we can efficiently estimate the expected distance from q to its nearest neighbor in the random subset? These basic computational geometry problems, whose complexity is quite well-understood in the deterministic setting, prove to be surprisingly hard in our stochastic setting. We obtain hardness results and approximation algorithms for stochastic problems of this kind.
computational geometry approximation algorithms probabilistic optimization data structures