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Abstract.We describe an n-processor, O(log(n) log log(n))-time CRCW algorithm to construct the Voronoi diagram for a set of n point-sites in the plane.Key Words. Voronoi diagram, Parallel algorithm.1. Introduction. Outline of the Algorithm. The Voronoi diagram is a geometric structure of great computational interest: see [5] for a useful survey. This paper addresses the problem of constructing the diagram in parallel, given as input a set of n points ("sites") in the plane. The Voronoi diagram for a set of sites is the locus of points equidistant from two closest sites: Figure 1 illustrates a diagram with 32 sites.The model of parallelism we assume is a CRCW PRAM, a system of independent processors accessing a shared random-access memory, where the same memory cell can be read by several processors simultaneously (concurrent read) and written by several processors simultaneously (concurrent write). Write-conflicts are resolved arbitrarily: the model of computation is an ARBITRARY CRCW PRAM.Each processor is assumed capable of exact rational and integer arithmetic in unit time.Earlier algorithms [1], [9] were presented for CREW 5 machines. The algorithm in [1] used n processors and took O(log 2 (n)) parallel time; that in [2] used n log(n) processors and took O(log(n) log log(n)) parallel time. Our algorithm reduces the overall work (parallel time × number of processors) to O(n log(n) log log(n)), while maintaining a runtime of O(log(n) log log(n)). Both of these figures are within the factor log log(n)

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This paper presents a worst-case optimal algorithm for constructing the Voronoi diagram for n disjoint convex and rounded semi-algebraic sites in 3 dimensions.Rather than extending optimal 2-dimensional methods, 32 ' 16 ' 20 ' 2 we base our method on a suboptimal 2-dimensional algorithm, outlined by Lee and Drysdale and modified by Sharir 20 ' 30 for computing the diagram of circular sites.For complexity considerations, we assume the sites have bounded complexity, i.e., the algebraic degree is bounded as is the number of algebraic patches making up the site. For the sake of simplicity we assume that the sites are what we call rounded. This assumption simplifies the analysis, though it is probably unnecessary.Our algorithm runs in time 0(C(n)) where C(n) is the worst-case complexity of an n-site diagram. For spherical sites C(n) is 9(n 2 ), but sharp estimates do not seem to be available for other classes of site.

Whittlesey 12 > 13 > 14 gave a criterion which decides when two finite 2-dimensional complexes are homeomorphic. We show that graph isomorphism can be reduced efficiently to 2-complex homeomorphism, and that Whittlesey's criterion can be reduced efficiently to graph isomorphism. Therefore graph isomorphism and 2-complex homeomorphism are polynomial-time equivalent.