Computational Geometry
A Parallel algorithm for Delaunay Triangulation
Friday, July 13, 2001
Mantena V. Raju
Department of Computing and Information Sciences, KSU
[email protected]
Kansas State University
Department of Computing and Information Sciences
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.
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(n2)
Kansas State University
Department of Computing and Information Sciences
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
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/
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.
Kansas State University
Department of Computing and Information Sciences
Parallel Delaunay Triangulation
Kansas State University
Department of Computing and Information Sciences