Simultaneous Nearest Neighbor Search

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 = q1, · · · , qk, and a compatibility graph G with vertices in Q, and the goal is to return data points p1, · · · , pk that minimize (i) the weighted sum of the distances from qi to pi and (ii) the weighted sum, over all edges (i, j) in the compatibility graph G, of the distances between pi and pj . 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 qi we find its (approximate) nearest neighbor point p̂i; this can be done efficiently using existing approximate nearest neighbor structures. In the second step, we solve an off-line optimization problem over sets q1, · · · , qk and p̂1, · · · , p̂k; this can be done efficiently given that k is much smaller than n. Even though p̂1, · · · , p̂k might not constitute the optimal answers to queries q1, · · · , qk, 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.