Closest pair and the post office problem for stochastic points

Authors:
Pegah Kamousi;Timothy M. Chan;Subhash Suri
Affiliations:
Department of Computer Science, UC Santa Barbara, CA 93106, USA;Cheriton School of Computer Science, University of Waterloo, Ontario N2L 3G1, Canada;Department of Computer Science, UC Santa Barbara, CA 93106, USA
Venue:
Computational Geometry: Theory and Applications
Year:
2014

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Abstract

Given a (master) set M of n points in d-dimensional Euclidean space, consider drawing a random subset that includes each point m"i@?M with an independent probability 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.