COMPUTATIONAL GEOMETRY -- VORONOI DIAGRAM

1. COMPUTATIONAL GEOMETRY-- VORONOI DIAGRAM Sophie Che

2. THE LARGEST EMPTY CIRCLE PROBLEM An application of Voronoi Diagram

4. S : site points v : Voronoi vertex v Largest Empty Circle e : Voronoi edge e

5. Definitions &Basic Properties Of Voronoi Diagram

6. pi Definition

7. FORMAL DEFINITION • Let P be a set of n distinct points (sites) in the plane. • The Voronoi diagram of P is the subdivision of the plane into n cells, one for each site. • A point q lies in the cell corresponding to a site pi P iff Euclidean_Distance(q, pi) < Euclidean_distance(q, pj) for each pi P, j  i.

8. STRUCTURE • 2 sites • Perpendicular Bisector • 3 sites • Non-collinear sites • Half lines meeting at a Voronoi vertex • 4 sites • A bounded cell

9. pi STRUCTURE (CONT’D) • More sites… V(pi) = 1 j  n, j  i h(pi,pj)

10. COMPLEXITY (SIZE) • Intuitively: n2 • n sites • At most n-1 vertices and edges per cell • Actually: O(n) • For n 3, |v|  2n  5, |e|  3n  6

11. LINEAR COMPLEXITY – PROOF • Euler’s Formula: • For planar graphs, |V| – |E| + F = 2 • Connecting all half-infinite edges to an extra vertex  (|v| + 1) – |e| + n = 2 • Plus,  |v|  2n  5 |e|  3n  6 p e pi

12. Delaunay Triangulation Dual graph of Voronoi Diagram

13. Delaunay Triangulation • Dual graph • Nodes: site points • Edges: “neighboring” sites in VD • Properties • Planar graph • Linear complexity

14. Delaunay Properties • DT is the straight-line dual of VD. • DT is a triangulation if no four sites are cocircular. • Each face of DT corresponds to a vertex of VD. • Each edge of DT corresponds to an edge of VD. • Each node of DT corresponds to a cell of VD. • The boundary of DT is the convex hull of the sites. • The interior of each face of DT contains no sites.

15. Voronoi Properties • Each Voronoi cell is convex. • The cell of pi is unbounded iffpi is on the convex hull of the sites. [c.f. D6] • If v is a Voronoi vertex at the junction of p1, p2 and p3, then v is the center of the circle C(v) determined by p1, p2 and p3. • C(v) is the circumcircle for the Delaunay triangle corresponding to v. • The interior of C(v) contains no sites. [c.f. D7] • If pi is a nearest neighbor to pj, then pipj is an edge of DT. • If there is some circle through pi and pj that contains no other sites, then pipj is an edge of DT.

16. VORONOI VERTEX • A Voronoi vertex is an intersection of 3 (or more) segments, each equidistant from a pair of sites • A point q is a Voronoi vertex iff its largest empty circle centered at q touches at least 3 (or more) sites pk pi q pj

17. VORONOI EDGE • A Voronoi edge is a subset of locus of points equidistant from pi and pj • A point q lies on a Voronoi edge between sites pi and pjiffits largest empty circle centered at q touches onlypi and pj pi pj q

18. Algorithms To compute a Voronoi Diagram

19. COMPUTING VORONOI DIAGRAM

20. HALF-PLANE INTERSECTION • For each site pi, Compute the common intersection of the half-planes h(pi, pj), with j≠i

21. HALF-PLANE INTERSECTION • For each site pi, Compute the common intersection of the half-planes h(pi, pj), with j≠i

22. HALF-PLANE INTERSECTION • For each site pi, Compute the common intersection of the half-planes h(pi, pj), with j≠i • O(nlogn) time per Voronoi cell • O( n2 log n ) in total

23. INCREMENTAL CONSTRUCTION • Incremental insertion • Obtain VDn from VDn-1 • Steps: • Locate pi in some Voronoi cell of pj; • Draw bisector between piand pj; • Expand boundary of Voronoi cell of pi; pj pi

24. INCREMENTAL CONSTRUCTION • Incremental insertion • Obtain VDn from VDn-1 • Steps: • Locate pi in some Voronoi cell of pj; • Draw bisector between piand pj; • Expand boundary of Voronoi cell of pi; pj pi

25. INCREMENTAL CONSTRUCTION • Incremental insertion • Obtain VDn from VDn-1 • Steps: • Locate pi in some Voronoi cell of pj; • Draw bisector between piand pj; • Expand boundary of Voronoi cell of pi; pj pi

26. INCREMENTAL CONSTRUCTION • Incremental insertion • Obtain VDn from VDn-1 • O(n) time per point insertion • O(n2) in total • Demo

27. GENERAL SWEEP LINE FRAMEWORK • Sweep a horizontal line (sweep line) from top to bottom over the plane • Incremental construction • Maintain unchangedinformation • Handle changes atcertain special points(event points) l

28. FORTUNE’S ALGORITHM • Problems: • What to maintain? • Intersections? • Beach line • unchanged information – Distance l

29. BEACH LINE • Sequence of Parabolic arcs Set of points equidistant from nearest site above sweep line & sweep line itself • breakpoints lie on Voronoi edges • How to store it?? • Sites consist of the line l breakpoint Equidistance arc

30. FORTUNE’S ALGORITHM • Demo A visual implementation of Fortune's Voronoi algorithm

31. EVENTS NEEDED TO BE HANDLED – SITE EVENT • When the sweep line reaches a new site… l

32. EVENTS NEEDED TO BE HANDLED – SITE EVENT • When the sweep line reaches a new site… l

33. EVENTS NEEDED TO BE HANDLED – SITE EVENT • When the sweep line reaches a new site…  A new arc appears l

34. HANDLING SITE EVENTS • Find where it projects on beach line • Insert its arc in proper position in the list of beach line

35. EVENTS NEEDED TO BE HANDLED – CIRCLE EVENT • When the sweep line encounters the bottom of an empty circle touching 3 or more sites… pi pj pk l

36. EVENTS NEEDED TO BE HANDLED – CIRCLE EVENT • When the sweep line encounters the bottom of an empty circle touching 3 or more sites…  An arc disappears pi pj pk y l

37. HANDLING CIRCLE EVENTS • Remove pj from beach line • Create a Voronoi vertex y • Center of empty circle is a Voronoi vertex • Tie two Voronoi edges into y • Create a Voronoi edge between pi and pk

38. RUNNING TIME ANALYSIS • How many arcs in beach line? • Each site split at most 1 existing arc into 2  At most 2n-1 arcs • How many events? • Site events • O(n) – number of sites • Circle events • O(n) – number of arcs

39. RUNNING TIME ANALYSIS – DATA STRUCTURE • Beach line – sites that consist of the line  A binary search tree • Event queue  A priority queue

40. RUNNING TIME ANALYSIS – SITE EVENT • Find where it projects on beach line • O(log n) • Insert its “arc” in proper position in the list of beach line • O(1)

41. RUNNING TIME ANALYSIS – CIRCLE EVENT • Remove pj from beach line • O(log n) • Create a Voronoi vertex y • O(1) • Tie two Voronoi edges into y • O(1) • Create a Voronoi edge between pi and pk • O(1)

42. TOTAL RUNNING TIME • Each site/circle event handler O(log n) • At most O(n) site and circle events  O(n log n) total running time

43. IS FORTUNE’S ALGORITHM OPTIMAL? • Sorting n real numbers is reducible to computing Voronoi diagrams  Lower bound O(n log n) • Fortune’s algorithm is optimal!

44. SUMMARY • VD is a useful planar subdivision of a discrete point set • VD has linear complexity/size • O(n) edges and O(n) vertices • Can be constructed in O(n log n) time • Fortune’s algorithm (optimal)

45. THANK YOU Questions ??

46. SPECIAL CASES Collinear sites Co-circle sites v

47. LINEAR COMPLEXITY – PROOF • Case 1: collinear |v| = 0, |e| = n – 1