An Improved Algorithm for the Fixed-Radius Neighbor Problem

The planar fixed-radius near neighbor problem can be stated as follows: Preprocess a set P of N points in the plane so that all points of P lying within some fixed radius r of a new point can be listed effectively. This problem has many practical applications in domains as varied as molecular graphics, statistics, air traffic control or data transmission [3]. Although a great deal of work has been done on the subject, the intricacy of the L, (Euclidean) metric has often led to consider the L, (Manhattan) or L, metrics instead (where the locus of points within a fixed distance of a given point becomes a square), or to introduce simplifying assumptions on the density of points (e.g., sparsity conditions) [3]. In [ 11, however, Bentley and Maurer do examine the worst-case performance of our searching problem in the Euclidean metric. and they develop a locus method which is optimal for that criterion. More precisely, for any query point M, the A points of the set P less than r away from M can be reported in A + log N time. Unfortunately, the preprocessing involved becomes rapidly prohibitive as N grows, since the time P(N)