Abstract
A well-known simple heuristic algorithm for solving the all-nearest-neighbors problem in thek-dimensional Euclidean spaceE k,k>1, projects the given point setS onto thex-axis. For each pointq εS a nearest neighbor inS under anyL p -metric (1 ≤p ≤ ∞) is found by sweeping fromq into two opposite directions along thex-axis. If δ q denotes the distance betweenq and its nearest neighbor inS the sweep process stops after all points in a vertical 2δ q -slice centered aroundq have been examined. We show that this algorithm solves the all-nearest-neighbors problem forn independent and uniformly distributed points in the unit cube [0,1]k in Θ(n 2−1/k) expected time, while its worst-case performance is Θ(n 2).
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Bartling, F., Hinrichs, K. Probabilistic analysis of an algorithm for solving thek-dimensional all-nearest-neighbors problem by projection. BIT 31, 558–565 (1991). https://doi.org/10.1007/BF01933171
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DOI: https://doi.org/10.1007/BF01933171