Fourth Msi Workshop on Computational

The Delaunay Triangulation (DT) is fundamental to computational geometry and arises naturallyin many domains. Sequential algorithms for DT have been studied from both a theoretical andpractical point of view. Fortune gives performance analysis of many of these algorithms [For92].Parallel algorithms for this problem have also been designed. There are several theoreticallyoptimal parallel algorithms, but they seem to have large constants in the work they perform and arealso complicated to code. Thus, of the the parallel implementation we are aware they are simpleralgorithms and efficient only for special data sets. In particular, they work best with uniformlydistributed data sets, [TSBP93, Su].We present a new parallel DT algorithm and its implementation. Our algorithm is based onthe “marriage before conquest”(pre-divide-and-conquer) paradigm. It has similarities to the 3-Dconvex hull algorithm of Edelsbrunner and Shi [ES91] and graph separator approach of Miller,Talmor and, Teng [MTT94]. We take advantage of the fact that there are parallel 2-D convexhull algorithms such as “quick hull”, another pre-divide-and-conquer parallel algorithm, that areefficient in practice.Computing the cut is done in the following way: We show that given any line L in the plane,we can compute the border C of a cut of the Delaunay Diagram, such that Delaunay triangleswhose circumcenter lies to the left(right) of L lie left(right) to C. The cut-border C is computedby mapping the points onto a 3-D parabola whose lowest point lies on the line L, projecting thepoints onto the vertical plane passing through L, and computing the 2D convex hull of the projectedpoints.We present an implementation of this algorithm in NESL. NESL is a data parallel languagebased on the segmented-scan vector model by Blelloch [Ble90], which currently runs on the CM-5,Cray C90 and workstations. Preliminary experiments indicate O(n log n) work. Our data setsincluded the following random distributions:(X;Y) = (W; Z) (X;Y) = (1=W;Z) (R; ) = (1=W; 2 Z)where W; Z are chosen uniform from (0;1). We found only a constant factor difference in the runtime depending on the distribution. The second two distributions are known not to work efficientlywith other algorithms[TSBP93, Su].We will also discuss dealing with degeneracies, and generalizations to 3D convex hull andDelaunay diagrams. We would like to give both a talk and a demo.46