Analysis of approximate nearest neighbor searching with clustered point sets

Nearest neighbor searching is a fundamental computational problem. A set of n data points is given in real d-dimensional space, and the problem is to preprocess these points into a data structure, so that given a query point, the nearest data point to the query point can be reported efficiently. Because data sets can be quite large, we are primarily interested in data structures that use only O(dn) storage. A popular class of data structures for nearest neighbor searching is the kd-tree and variants based on hierarchically decomposing space into rectangular cells. An important question in the construction of such data structures is the choice of a splitting method, which determines the dimension and splitting plane to be used at each stage of the decomposition. This choice of splitting method can have a significant influence on the efficiency of the data structure. This is especially true when data and query points are clustered in low dimensional subspaces. This is because clustering can lead to subdivisions in which cells have very high aspect ratios. We compare the well-known optimized kd-tree splitting method against two alternative splitting methods. The first, called the sliding-midpoint method, which attempts to balance the goals of producing subdivision cells of bounded aspect ratio, while not producing any empty cells. The second, called the minimum-ambiguity method is a query-based approach. In addition to the data points, it is also given a training set of query points for preprocessing. It employs a simple greedy algorithm to select the splitting plane that minimizes the average amount of ambiguity in the choice of the nearest neighbor for the training points. We provide an empirical analysis comparing these two methods against the optimized kd-tree construction for a number of synthetically generated data and query sets. We demonstrate that for clustered data and query sets, these algorithms can provide significant improvements over the standard kd-tree construction for approximate nearest neighbor searching. 1991 Mathematics Subject Classification. 68P10, 68W40.