Voronoi Diagrams. A Voronoi diagram records everything one would ever want to know about proximity to a set of points

Transcription

1 Voronoi Diagrams

2 Voronoi Diagrams A Voronoi diagram records everything one would ever want to know about proximity to a set of points Who is closest to whom? Who is furthest? We will start with a series of examples

3 Application: Preview Fire Observation Towers Towers on Fire Nearest Neighbor Clustering Facility Location Path Planning Crystallography

4 Medial Axis

5 Voronoi Diagrams Definitions and Basic Properties

6 Definitions and Basic Properties Input: Let P = {p 1, p 2,, p n } be a set of 2-d points Partition the plane by assigning every point in the plane to its nearest site

7 Definitions and Basic Properties

8 Definitions and Basic Properties

9 Definitions and Basic Properties Halfplanes Let H (p i, p j ) be the closed halfplane with boundary B ij and containing p i Then, H(p i, p j ) can be viewed as all the points that are closer to p i than they are to p j Recall that V(p i ) is the set of all points closer to p i than to any other site

10 Definitions and Basic Properties Four Sites

11 Definitions and Basic Properties Many Sites

12 Delaunay Triangulations

13 Properties V1 Each Voronoi region V(p i ) is convex V2 V(p i ) is unbounded iff p i is on the convex hull of the point set V3 If v is a Voronoi vertex at the junction of V(p 1 ), V(p 2 ), and V(p 3 ), then v is the center of the circle C(v) determined by p 1, p 2, and p 3 V4 C(v) is the circumcircle for the Delaunay triangle corresponding to v

14 Properties V5 The interior of C(v) contains no sites V6 If p j is a nearest neighbor to p i, then (p i, p j ) is an edge of D(P) V7 If there is some circle through p i and p j that contains no other sites, then (p i, p j ) is an edge of D(P). The reverse also holds: For every Delaunay edge, there is some empty circle

15 Largest Empty Circle

16 Definitions and Basic Properties Complexity of the Voronoi Diagram?

17 Properties of Delaunay Triangulations Delaunay triangulation and Voronoi diagram are dual structures each contains the same information in some sense but represented in a rather different form D1 D(P) is the straight-line dual of V(P) D2 D(P) is a triangulation if no four points of P are cocircular: Every face is a triangle. The faces of D(P)are called Delaunay triangles D3 Each face (triangle) of D(P) corresponds to a vertex of V(P)

18 Properties of Delaunay Triangulations 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 (triangle) face of D(P) contains no sites. (Compare V5.)

19 Voronoi Diagrams Algorithms

20 Intersection of Halfplanes Constructing the intersection of n halfplanes is dual to the task of constructing the convex hull of n points in two dimensions can be accomplished with similar algorithms in O(nlogn) time. Doing this for each site would cost O(n 2 logn)

21 Incremental Construction Suppose the Voronoi diagram V for k points is already constructed, and now we would like to construct the diagram V after adding one more point p Suppose p falls inside the circles associated with several Voronoi vertices, say C(v 1 ),, C(v m ). Then these vertices of V cannot be vertices of V, because of V5 These are the only vertices of V that are not carried over to V These vertices are also all localized to one area of the diagram The algorithm spends O(n) time per point insertion, for a total complexity of O(n 2 )

22 Divide and Conquer O(nlogn) time-algorithm First detailed by Shamos & Hoey (1975) This time complexity is asymptotically optimal, but the algorithm is rather difficult to implement Careful attention to data structures required

23 Fortune s Algorithm

24 Fortune s Algorithm

25 Voronoi Diagrams Connection to Convex Hulls

26 Two-Dimensional Delaunay Triangulations New we repeat the same analysis in two dimensions The paraboloid is z = x 2 + y 2, see Figure 5.24

27 Two-Dimensional Delaunay Triangulations

28 Two-Dimensional Delaunay Triangulations

29 Assignment EX: (due Oct 22) Quiz next week! 29