Simultaneous nearest neighbor search

Abstract

Motivated by applications in computer vision and databases, we introduce and study the Simultaneous Nearest Neighbor Search (SNN) problem. Given a set of data points, the goal of SNN is to design a data structure that, given a collection of queries, finds a collection of close points that are compatible with each other. Formally, we are given k query points Q=q_1,...,q_k, and a compatibility graph G with vertices in Q, and the goal is to return data points p_1,...,p_k that minimize (i) the weighted sum of the distances from q_i to p_i and (ii) the weighted sum, over all edges (i,j) in the compatibility graph G, of the distances between p_i and p_j. The problem has several applications in computer vision and databases, where one wants to return a set of *consistent* answers to multiple related queries. Furthermore, it generalizes several well-studied computational problems, including Nearest Neighbor Search, Aggregate Nearest Neighbor Search and the 0-extension problem. In this paper we propose and analyze the following general two-step method for designing efficient data structures for SNN. In the first step, for each query point q_i we find its (approximate) nearest neighbor point p'_i; this can be done efficiently using existing approximate nearest neighbor structures. In the second step, we solve an off-line optimization problem over sets q_1,...,q_k and p'_1,...,p'_k; this can be done efficiently given that k is much smaller than n. Even though p'_1,...,p'_k might not constitute the optimal answers to queries q_1,...,q_k, we show that, for the unweighted case, the resulting algorithm satisfies a O(log k/log log k)-approximation guarantee. Furthermore, we show that the approximation factor can be in fact reduced to a constant for compatibility graphs frequently occurring in practice, e.g., 2D grids, 3D grids or planar graphs. Finally, we validate our theoretical results by preliminary experiments. In particular, we show that the empirical approximation factor provided by the above approach is very close to 1.