Chapter 5: Voronoi Diagrams Wednesday, 2/14/07

1. UMass Lowell Computer Science 91.504Advanced AlgorithmsComputational GeometryProf. Karen DanielsSpring, 2007 Chapter 5: Voronoi Diagrams Wednesday, 2/14/07

2. Chapter 5Voronoi Diagrams Applications Preview Definitions & Basic Properties Delaunay Triangulations Algorithms Applications in Detail Medial Axis Connection to Convex Hulls Connection to Arrangements

3. Applications Preview • Nearest neighbors • Minimum spanning tree • Medial axis http://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html

4. Convex Hull New Point Delaunay Triangulation Common Computational Geometry Structures Voronoi Diagram Combinatorial size in O(n)

5. Basics • Voronoi region V(pi) is set of all points at least as close to pi as to any other site:

6. Delaunay Triangulations: Properties http://www.cs.cornell.edu/Info/People/chew/Delaunay.html

7. Delaunay Triangulations: Properties • D1. D(P) is the straight-line dual of V(P) [by definition] • D2. D(P) is a triangulation if no 4 points of P are cocircular: Every face is a triangle. • D3. Each face of D(P) corresponds to a vertex of V(P) • D4. Each edge of D(P) corresponds to an edge of V(P) • D5. Each node of D(P) corresponds to a region of V(P) • D6. The boundary of D(P) is the convex hull of the sites • D7. The interior of each face of D(P) contains no sites

8. Delaunay Triangulations: Properties of Voronoi Diagrams http://www.cs.cornell.edu/Info/People/chew/Delaunay.html

9. Voronoi Diagram: Properties • V1. Each Voronoi region V(pi) is convex • V2. V(pi) is unbounded iff pi is on convex hull of point set • V3. If v is a Voronoi vertex at junction of V(p1), V(p2), V(p3), then v is center of circle C(v) determined by p1, p2, and p3. • V4. C(v) is circumcircle for Delaunay triangle for v • V5. Interior(C(v)) contains no sites • V6. If pi is a nearest neighbor to pj, then (pi, pj) is an edge of D(P) • V7. If there is some circle through pi and pj that contains no other sites, then (pi, pj) is an edge of D(P). [reverse holds too]

10. Algorithms • Half-Plane Intersection: • Construct each Voronoi region separately: • Intersect n-1 halfplanes • O(n lg n) time • Total time: O(n2 lg n) • Incremental Construction: • Add one point at a time • If new point is inside some Voronoi circle, locally update diagram • O(n) work per point • Total time: O(n2) • Divide-and-Conquer: O(n lg n)

11. 45 45 slope Algorithms: Fortune • Plane-sweep complication: What if encounter Voronoi edges before corresponding site? • Solution: • Expanding circles interpretation (z is time dimension) • 2 cones intersect in branch of hyperbola that projects down onto (potential) Voronoi edge • If cones opaque, then Voronoi diagram is view from z = • Sweep-line is intersection of slanted plane p with xy-plane • Because p slanted at same angle as cones, L hits site when p first hits cone for site • Invariant: Everything visible underneath p is always finished (that part of Voronoi diagram is done) • Maintain parabolic front = projection of intersection of p with cones onto xy-plane • Events • Site event : new (degenerate, single-line) parabola forms. 2 (duplicate) intersection points with parabolic front form new degenerate Voronoi diagram edge • Circle event: 2 parabolic front intersection points meet to form Voronoi vertex. • As parabolic front evolves, parabola intersection points trace out Voronoi diagram. p x O(nlgn) time sweep line L at x = l

12. How are they related?Delaunay Triangulation • Project each point upwards onto paraboloid z= x2 + y2 • Construct 3D Convex Hull • Discard “top” faces • Project Convex Hull down to xy plane to form Delaunay Triangulation • View from z = -infinity to see Delaunay Triangulation

13. How are they related?Voronoi Diagram • Project each point upwards onto paraboloid z= x2 + y2 • At each projected point, construct plane tangent to paraboloid • tangent above (a,b) is • z = 2ax + 2by - (a2 + b2) • View from z = +infinity to see Voronoi Diagram • “first intersection” of planes

14. Applications: Nearest Neighbors Nearest Neighborhood query for point q with respect to points P = p1, p2 ,..., pn can be answered using Voronoi diagram All Nearest Neighbors: The Nearest Neighborhood Graph (NNG) of points Pis a graph: -whose nodes correspond to the points - an arc from pi to pj iff pj is a nearest neighbor of pi :

15. Applications: Minimum Spanning Tree MST can be used in TSP approximation algorithm.

16. LUNE forbidden region Applications: Relative Neighborhood Graph The Relative Neighborhood Graph (RNG) of points p1, p2 ,..., pn is a graph: -whose nodes correspond to the points - 2 nodes p1, p2 share an arc iff they are at least as close to each other as to any other point: Note: this produces a set of constraints (1 for each m) It can be shown that: - every edge of the RNG is an edge of the Delaunay triangulation - every edge of an MST is an edge of the RNG RNG for 4 sample points

17. Applications: Largest Empty Circle Problem: Find a largest empty circle whose center is in the (closed) convex hull of a set of n sites S (empty in that it contains no sites in its interior, and largest in that there is no other such circle with strictly larger radius). Finite set of potential largest empty circle centers: - Voronoi vertices - Intersections of Voronoi edges and hull of sites.

18. Applications: Medial Axis Medial Axis Pruning: Robert Ogniewicz of Harvard uses medial axes for shape recognition. http://www.ics.uci.edu/~eppstein/gina/medial.html

19. Applications: Medial Axis • Medial Axis: of polygon P = set of points inside P that have > 1 closest point among P’s boundary points. • Tree whose leaves are vertices of P • Every point of medial axis is center of circle touching boundary in at least 2 points. • Vertices of medial axis are centers of circles touching 3 distinct boundary points. • “Grassfire” analogy: fire starts at boundary • Medial axis represents “quench points” where fires meet each other

20. Applications: Shewchuck Triangulation http://www.cs.cmu.edu/~quake/triangle.demo.html