Computational geometry: an introduction
Computational geometry: an introduction
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Constrained Delaunay triangulations
SCG '87 Proceedings of the third annual symposium on Computational geometry
An optimal algorithm for constructing the Delaunay triangulation of a set of line segments
SCG '87 Proceedings of the third annual symposium on Computational geometry
Cascading divide-and-conquer: a technique for designing parallel algorithms
SIAM Journal on Computing
A linear-time algorithm for computing the Voronoi diagram of a convex polygon
Discrete & Computational Geometry
Parallel construction of subdivision hierarchies
Journal of Computer and System Sciences
Parallel transitive closure and point location in planar structures
SIAM Journal on Computing
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Optimal parallel randomized algorithms for three-dimensional convex hulls and related problems
SIAM Journal on Computing
An NC parallel 3D convex hull algorithm
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
New Parallel Algorithms for Convex Hull and Triangulation in 3-Dimensional Space
MFCS '92 Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science
Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Parallel algorithms for geometric problems
Parallel algorithms for geometric problems
Parallel algorithms for higher-dimensional convex hulls
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
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In this paper, we present an O(1/αlog n) (for any constant 0 ≤ α ≤ 1) time parallel algorithm for constructing the constrained Voronoi diagram of a set L of n non-crossing line segments in E2, using O(n1+α) processors on a CREW PRAM model. This parallel algorithm also constructs the constrained Delaunay triangulation of L in the same time and processor bound by the duality. Our method established the conversions from finding the constrained Voronoi diagram L to finding the Voronoi diagram of S, the endpoint set of L. We further showed that this conversion can be done in O(log n) time using n processors in CREW PRAM model. The complexity of the conversion implies that any improvement of the complexity for finding the Voronoi diagram of a point set will automatically bring the improvement of the one in question.