An optimal algorithm for the angle-restricted all nearest neighbor problem on the reconfigurable mesh, with applications

Given a set S of n points in the plane and two directions r₁ and r₂, the Angle-Restricted All Nearest Neighbor problem (ARANN, for short) asks to compute, for every point p in S, the nearest point in S lying in the planar region bounded by two rays in the directions r₁ and r₂ emanating from p. The ARANN problem generalizes the well-known ANN problem and finds applications to pattern recognition, image processing, and computational morphology. Our main contribution is to present an algorithm that solves an instance of size n of the ARANN problem in O(1) time on a reconfigurable mesh of size n×n. Our algorithm is optimal in the sense that Ω(n²) processors are necessary to solve the ARANN problem in O(1) time. By using our ARANN algorithm, we can provide O(1) time solutions to the tasks of constructing the Geographic Neighborhood Graph and the Relative Neighborhood Graph of n points in the plane on a reconfigurable mesh of size n×n. We also show that, on a somewhat stronger reconfigurable mesh of size n×n², the Euclidean Minimum Spanning Tree of n points can be computed in O(1) time