Parallel Solutions to Geometric Problems in the Scan Model of Computation

This paper describes several parallel algorithms that solve geometric problems. The algorithms are based on a vector model of computation-the scan model. The purpose of this paper is both to show how the model can be used and to formulate a set of practical algorithms. The scan model is based on a small set of operations on vectors of atomic values. It differs from the P-RAM models both in that it includes a set of scan primitives, also called parallel prefix computations, and in that it is a strictly data-parallel model. A very useful abstraction in the scan model is the segment abstraction, the subdivision of a vector into a collection of independent smaller vectors. The segment abstraction permits a clean formulation of divide-and-conquer algorithms and is used heavily in the algorithms described in this paper. Within the scan model, using the operations and routines defined, the paper describes a k-D tree algorithm requiring O(lg n) calls to the primitives for n points, a closest-pair algorithm requiring O(lg n) calls to the primitives, a line-drawing algorithm requiring O(1) calls to the primitives, a line-of-sight algorithm requiring O(1) calls to the primitives, and finally, three different convex-hull algorithms. The last convex-hull algorithm, merge-hull, utilizes a generalized binary search technique using divide-and-conquer with the segment abstraction. The paper also describes how to implement the CREW version of Cole's merge sort in O(lg n) calls to the primitives. All these algorithms should be noted for their simplicity rather than their complexity; many of them are parallel versions of known serial algorithms. Most of the algorithms discussed in this paper have been implemented on the Connection Machine, a highly parallel single instruction multiple data computer.