Asymptotically efficient hypercube algorithms for computational geometry

Abstract

Hypercube algorithms that solve many fundamental computational geometry problems are presented. The algorithms use decomposition techniques, which enable them to outperform asymptotically the fastest previous algorithms for these problems. Previous algorithms all run in Theta (log/sup 2/n) time, even when using a sorting method that runs in o(log/sup 2/n) time. The new algorithms use a recently discovered o(log/sup 2/n) time sorting method to improve their asymptotic speed to o(log/sup 2/n). If sorting runs in Theta (Sort(n)) time, the algorithms for two-set dominance counting, 3-D maxima, closest pair, and all points nearest neighbors run in Theta (Sort(n)) log(log n) time, and the algorithms for triangulation and visibility from a point run in Theta (Sort(n)) time.<>