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<p><b>Abstract</b>—Given a set <it>S</it> of <it>n</it> points in the plane and two directions <tmath>$r_1$</tmath> and <tmath>$r_2,$</tmath> the Angle-Restricted All Nearest Neighbor problem (ARANN, for short) asks to compute, for every point <it>p</it> in <it>S</it>, the nearest point in <it>S</it> lying in the planar region bounded by two rays in the directions <tmath>$r_1$</tmath> and <tmath>$r_2$</tmath> emanating from <it>p</it>. 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 <it>n</it> of the ARANN problem in <it>O</it>(1) time on a reconfigurable mesh of size <it>n</it>×<it>n</it>. Our algorithm is optimal in the sense that <tmath>$\Omega\;(n^2)$</tmath> processors are necessary to solve the ARANN problem in <it>O</it>(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 <it>n</it> points in the plane on a reconfigurable mesh of size <it>n</it>×<it>n</it>. We also show that, on a somewhat stronger reconfigurable mesh of size <tmath>$n\times n^2,$</tmath> the Euclidean Minimum Spanning Tree of <it>n</it> points can be computed in <it>O</it>(1) time.</p>
Reconfigurable mesh, lower bounds, proximity problems, ANN, ARANN, mobile computing.

K. Nakano and S. Olariu, "An Optimal Algorithm for the Angle-Restricted All Nearest Neighbor Problem on the Reconfigurable Mesh, with Applications," in IEEE Transactions on Parallel & Distributed Systems, vol. 8, no. , pp. 983-990, 1997.
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