Optimal CREW-PRAM algorithms for direct dominance problems

Abstract

We present optimal parallel solutions to direct dominance problems for planar point sets. Our algorithms are deterministic and designed to run on the concurrent read exclusive write parallel random-access machine (CREW PRAM). In particular, we provide algorithms for counting the number of points that are directly dominated by each point of a planar point set, and for reporting these point sets. The counting algorithm runs in O(log n) time using O(n) processors; the reporting algorithm runs in O(log n) time using O(n+k/log n) processors, where k is the size of the output. The total work of each algorithm matches the respective sequential lower bound. As an application of our results, we present an algorithm for the maximum empty rectangle problem, which is work optimal in the expected case.