Improved parallel bucketing algorithms for proximity problems

A new parallel algorithm for constructing Voronoi diagrams is presented. For n input sites drawn independently from the uniform distribution over the unit square, the expected running time is O(log*n) on the Tolerant parallel random access machine (PRAM) and O(1) on the stronger OR PRAM (log*n=1) for n⩽2, and log*n=1+log*(log ₂n for n>2). In both cases the number of processors used is optimal for the given running time, i.e, O( n/log*n) for the Tolerant PRAM and O(n) for the OR PRAM, and the probability that the running time exceeds its expected value by more than a constant factor is exponentially small. The algorithm is based on the bucketing approach. Using similar techniques, other closely related proximity problems are shown to be solvable with the same resource bounds