Efficient parallel solutions to some geometric problems
Journal of Parallel and Distributed Computing
A note on relative neighborhood graphs
SCG '87 Proceedings of the third annual symposium on Computational geometry
Parallel computational geometry
Parallel computational geometry
The crust and the &Bgr;-Skeleton: combinatorial curve reconstruction
Graphical Models and Image Processing
The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees
Journal of the ACM (JACM)
Beta-skeletons have unbounded dilation
Computational Geometry: Theory and Applications
Fast Algorithms for Computing beta-Skeletons and Their Relatives
ISAAC '97 Proceedings of the 8th International Symposium on Algorithms and Computation
Fast Construction of k-Nearest Neighbor Graphs for Point Clouds
IEEE Transactions on Visualization and Computer Graphics
On the shape of a set of points in the plane
IEEE Transactions on Information Theory
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β-skeletons, a prominent member of the neighborhood graph family, have interesting geometric properties and various applications ranging from geographic networks to archeology. This paper focuses on computing the β-spectrum, a labeling of the edges of the Delaunay Triangulation, DT(V), which makes it possible to quickly find the lune-based β-skeleton of V for any query value β∈[1, 2]. We consider planar n point sets V with Lp metric, 1pO(n log2n) time sequential, and a O(log4n) time parallel β-spectrum labeling. We also show a parallel algorithm, which for a given β∈[1,2], finds the lune-based β-skeleton in O(log2n) time. The parallel algorithms use O(n) processors in the CREW-PRAM model.