Stochastic closest-pair problem and most-likely nearest-neighbor search in tree spaces

TL;DR

This paper introduces algorithms for stochastic closest-pair and nearest-neighbor problems in tree spaces, providing efficient computation of probabilities, expectations, and data structures for approximate nearest-neighbor queries.

Contribution

It presents the first algorithms for computing threshold probabilities and expected distances in stochastic tree spaces, and develops a $k$-most-likely Voronoi diagram for efficient nearest-neighbor search.

Findings

First algorithm for $ ext{l}$-threshold probability in $O(t+n ext{log}n+ ext{min}\{tn,n^2 ight)$ time.

Efficient approximation of expected closest-pair distance within $O(t+ ext{epsilon}^{-1} ext{min}\{tn^2,n^3 ight)$ time.

Data structures for $k$-LNN queries with $O( ext{log}n+k)$ query time and $O(t+k^2n)$ average-case space.

Abstract

Let $T$ be a tree space (or tree network) represented by a weighted tree with $t$ vertices, and $S$ be a set of $n$ stochastic points in $T$, each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the $\ell$-threshold probability and the expectation of the closest-pair distance of a realization of $S$. We propose the first algorithm to compute the $\ell$-threshold probability in $O(t+n\log n+ \min\{tn,n^2\})$ time for any given threshold $\ell$, which immediately results in an $O(t+\min\{tn^3,n^4\})$-time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a $(1+\varepsilon)$-approximation for the expected closest-pair distance in $O(t+\varepsilon^{-1}\min\{tn^2,n^3\})$ time, by arguing that the expected closest-pair distance can be approximated via $O(\varepsilon^{-1}n)$ threshold probability queries. For the latter, we study the $k$ most-likely nearest-neighbor search ($k$-LNN) via a notion called $k$ most-likely Voronoi Diagram ($k$-LVD). We show that the size of the $k$-LVD $\varPsi_T^S$ of $S$ on $T$ is bounded by $O(kn)$ if the existence probabilities of the points in $S$ are constant-far from 0. Furthermore, we establish an $O(kn)$ average-case upper bound for the size of $\varPsi_T^S$, by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers $k$-LNN queries in $O(\log n+k)$ time using average-case $O(t+k^2n)$ space, and worst-case $O(t+kn^2)$ space if the existence probabilities are constant-far from 0. Finally, we also give an $O(t+ n^2\log n+n^2k)$-time algorithm to construct the LVD data structure.