On the Most Likely Voronoi Diagramand Nearest Neighbor Searching

Abstract

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.

The authors gratefully acknowledge support from the National Science Foundation, under the grants CNS-1035917 and CCF-11611495, and DARPA.