Chromatic K-Nearest Neighbor Queries

Let P be a set of n colored points in R ^d. We develop efficient data structures that store P and can answer chromatic k-nearest neighbor (k-NN) queries. Such a query consists of a query point q and a number k, and asks for the color that appears most frequently among the k points in P closest to q. Answering such queries efficiently is the key to obtain fast k-NN classifiers. Our main aim is to obtain query times that are independent of k while using near-linear space. We show that this is possible using a combination of two data structures. The first data structure allow us to compute a region containing exactly the k-nearest neighbors of a query point q, and the second data structure can then report the most frequent color in such a region. This leads to linear-space data structures with query times of (Formula Presented) for points in R ¹, and with query times varying between (Formula Presented) and O(n ^5/6 polylog n), depending on the distance measure used, for points in R ². Since these query times are still fairly large we also consider approximations. If we are allowed to report a color that appears at least (1 − ε)f ^*times, where f ^∗ is the frequency of the most frequent color, we obtain a query time of (Formula Presented) and expected query times ranging between (Formula Presented) and (Formula Presented) in R ² using near-linear space (ignoring polylogarithmic factors). All of our data structures are for the pointer-machine model.