On the Most Likely Voronoi Diagram and Nearest Neighbor Searching

We consider the problem of nearest-neighbor searching among a set of stochastic sites, where a stochastic site is a tuple \((s_i, \pi _i)\) consisting of a point \(s_i\) in a \(d\)-dimensional space and a probability \(\pi _i\) determining its existence. The problem is interesting and non-trivial even in \(1\)-dimension, where the Most Likely Voronoi Diagram (LVD) is shown to have worst-case complexity \(\Omega (n^2)\). We then show that under more natural and less adversarial conditions, the size of the \(1\)-dimensional LVD is significantly smaller: (1) \(\Theta (k n)\) if the input has only \(k\) distinct probability values, (2) \(O(n \log n)\) on average, and (3) \(O(n \sqrt{n})\) under smoothed analysis. We also present an alternative approach to the most likely nearest neighbor (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 worst-case with a bounded number of distinct probabilities. Using the Pareto-set approach, we can also reduce the multi-dimensional LNN search to a sequence of nearest neighbor and spherical range queries.