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

2. Kansas State University Department of Computing and Information Sciences Basic Definitions Convex Hull Given k distinct points P 1,P 2,…….. P k in E d the set of points P = a 1 P 1 + a 2 P 2 + ………+ a k P k where a 1,a 2,…….. a k are all  0 and a 1 + a 2 + ……..+ a k = 1 is the convex set generated by P 1,P 2,… P k and P is a convex combination of P 1,P 2,…….. P k. Example If P 1, P 2 are two points in E d, the convex combination of P 1, P 2 is the line segment joining P 1, P 2 Given an arbitrary subset L of points in E d the convex hull conv(L) of L is the smallest convex set containing L.

3. Kansas State University Department of Computing and Information Sciences Convex Hull Convex Hull of the points shown in blue dots Some algorithms for calculating Convex Hull in 2 dimensions 1.Graham’s Scan O(n log n) worst case running time. 2.Jarvis March O(nh) where h is the number of vertices of the convex hull. The worst case running time is O(n 2 )

4. Kansas State University Department of Computing and Information Sciences Voronoi Diagram Distance between 2 points p, q denoted by dist(p,q) = sqrt((p x - q x ) 2 + (p y - q y ) 2 Voronoi Diagram Let P = { P 1,P 2,…….. P n } 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 P i if and only if dist( P i,q) < dist( P j,q) for each P j  P

5. Kansas State University Department of Computing and Information Sciences Voronoi Diagram Some algorithms for calculating Voronoi in 2 dimensions 1.Fortunes’s Sweep line algorithm O(n log n) worst case running time using O(n) storage. 2.Guiba’s and Stolfi’s divide and conquer algorithm O(n log n) worst case running time. Animation of Fortune’s Sweep line Algorithm http://www.diku.dk/students/duff/Fortune/

6. Kansas State University Department of Computing and Information Sciences 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. Kansas State University Department of Computing and Information Sciences Parallel Delaunay Triangulation