A worst-case analysis of nearest neighbor searching by projection

Abstract

The nearest neighbor searching problem (also called the post office problem) calls for organizing the set P of N points in k-space so that the nearest neighbor in P to a new point can be quickly found. Friedman, Baskett and Shustek describe an algorithm for nearest neighbor searching based on projecting the points onto the various coordinate axes; their analysis of this method shows that a nearest neighbor search can be performed in O(N1−1/k) expected time, for any fixed dimension k>1 under a variety of probability distributions. In this paper we shall prove the stronger (worst-case) result that the total time required by (an extension of) their method to find the nearest neighbor of every point in any fixed k-dimensional point set is O(N2−1/k), which immediately implies a result similar to theirs. The above results hold only for the L∞ metric; we also investigate the Euclidean (L2) metric. Our first result for that metric shows that the above analysis does not hold in general, and our second result then goes on to show that the analysis does in fact apply in practice, because of the finite word-length restrictions of real computers.