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Friday, July 13, 2001 Mantena V. Raju Department of Computing and Information Sciences, KSU

Computational Geometry. A Parallel algorithm for Delaunay Triangulation. Friday, July 13, 2001 Mantena V. Raju Department of Computing and Information Sciences, KSU mra5577@cis.ksu.edu. Basic Definitions. Convex Hull

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Friday, July 13, 2001 Mantena V. Raju Department of Computing and Information Sciences, KSU

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  1. Computational Geometry A Parallel algorithm for Delaunay Triangulation Friday, July 13, 2001 Mantena V. Raju Department of Computing and Information Sciences, KSU mra5577@cis.ksu.edu

  2. Basic Definitions Convex Hull Given k distinct points P1,P2,……..Pk in Ed the set of points P = a1P1 + a2P2 + ………+akPk where a1,a2,……..ak are all  0 and a1 + a2+ ……..+ak = 1 is the convex set generated by P1,P2,…Pk and P is a convex combination of P1,P2,……..Pk. Example If P1, P2 are two points in Ed, the convex combination of P1, P2 is the line segment joining P1, P2 Given an arbitrary subset L of points in Ed the convex hull conv(L) of L is the smallest convex set containing L.

  3. Convex Hull Convex Hull of the points shown in blue dots Some algorithms for calculating Convex Hull in 2 dimensions • Graham’s ScanO(n log n)worst case running time. • Jarvis MarchO(nh) where h is the number of vertices of the convex hull. The worst case running time is O(n2)

  4. Voronoi Diagram Distance between 2 points p, q denoted by dist(p,q) = sqrt((px- qx)2 + (py- qy)2 Voronoi Diagram Let P = {P1,P2,……..Pn} be a set of n distinct points in the plane. Voronoi diagram of P is defined as the subdivsion of the plane into n cells, one for each site in P, with the property that a point q lies in the cell corresponding to a site Pi if and only if dist(Pi ,q) < dist(Pj ,q) for each Pj  P

  5. Voronoi Diagram Some algorithms for calculating Voronoi in 2 dimensions • Fortunes’s Sweep line algorithm O(n log n) worst case running time using O(n) storage. • Guiba’s and Stolfi’s divide and conquer algorithmO(n log n) worst case running time. • Animation of Fortune’s Sweep line Algorithm • http://www.diku.dk/students/duff/Fortune/

  6. Delaunay Triangulation Delaunay Triangulation is the straight line dual of the Voronoi Diagram. Algorithm for calculating Delaunay in 2 dimensions R. A. Dwyer’s Divide and Conquer O(n log log n) average case running time O(n log n) worst case running time.

  7. Parallel Delaunay Triangulation

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