Querying Probabilistic Neighborhoods in Spatial Data Sets Efficiently

Abstract

The probability that two spatial objects establish some kind of mutual connection often depends on their proximity. To formalize this concept, we define the notion of a probabilistic neighborhood: Let P be a set of n points in $\mathbb {R}^d$, $q \in \mathbb {R}^d$ a query point, ${\text {dist}}$ a distance metric, and $f : \mathbb {R}^+ \rightarrow [0,1]$ a monotonically decreasing function. Then, the probabilistic neighborhood N(q, f) of q with respect to f is a random subset of P and each point $p \in P$ belongs to N(q, f) with probability $f({\text {dist}}(p,q))$. Possible applications include query sampling and the simulation of probabilistic spreading phenomena, as well as other scenarios where the probability of a connection between two entities decreases with their distance. We present a fast, sublinear-time query algorithm to sample probabilistic neighborhoods from planar point sets. For certain distributions of planar P, we prove that our algorithm answers a query in $O((|N(q,f)| + \sqrt{n})\log n)$ time with high probability. In experiments this yields a speedup over pairwise distance probing of at least one order of magnitude, even for rather small data sets with $n=10^5$ and also for other point distributions not covered by the theoretical results.