On the Most Likely Voronoi Diagram and Nearest Neighbor Searching

Let 𝒮 = {(s1,π1), (s2,π2),…, (sn,πn)} be a set of stochastic sites, where each site is a tuple (si,πi) consisting of a point si in d-dimensional space and a probability πi of existence. Given a query point q, we define its most likely nearest neighbor (LNN) as the site with the largest probability of being q’s nearest neighbor. The Most Likely Voronoi Diagram (LVD) of 𝒮 is a partition of the space into regions with the same LNN. We investigate the complexity of LVD in one dimension and show that it can have size Ω(n2) in the worst-case. We then show that under non-adversarial conditions, the size of the 1-dimensional LVD is significantly smaller: (1) Θ(kn) if the input has only k distinct probability values, (2) O(nlog n) on average, and (3) O(nn) under smoothed analysis. We also describe a framework for LNN search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models as well as the worst-case with a bounded number of distinct probabilities. The Pareto-set framework is also applicable to multi-dimensional LNN search via reduction to a sequence of nearest neighbor and spherical range queries.

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