CHROMATIC K-NEAREST NEIGHBOR QUERIES

Let P be a set of a colored points in Rd. 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, and asks for the color that appears most frequently among the k points in P closest tolg. 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.<br /> 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 lincar-space data structures with query times of O(n(1)(/2)logn) for points in R-1, and with query times varying between O(n(2/3) log(2/3) n) 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)f" times, where f is the frequency of the most frequent color, we obtain a query time of O(log n + log log n) in (1) and expected query times ranging between O(n(1)/23/2) and O(n(1)/2-5/2) in R-2 using dear-linear space (ignoring polylogarithmic fac-tors). All of our data structures are for the pointer-machine model.