1. Order-k Voronoi diagram in the plane Dominique Schmitt Université de Haute-Alsace

2. (Very) Basic properties on circles

3. Voronoi diagram (reminder) V(s) : the set of points of the plane closer to s than to any other site = the Voronoi region of s Vor(S) : the partition of the plane formed by the Voronoi regions, their edges, and vertices S : a set of n sites in the plane

4. Voronoi regions, edges, and vertices V(s) : the set of centers of all the empty circles passing through s c(s,t) : the set of centers of all the empty circles passing through s and t u(Q) : the center of the empty circle passing through a set Q of at least 3 co-circular sites

5. Generalizations of the Voronoi diagram - d-dimensional space, - non-Euclidean space, - non-Euclidean distance, - weighted sites, - sites are not points, - k nearest neighbors -...

6. Farthest point Voronoi diagram V -1 (s) : the set of points of the plane farther from s than from any other site Vor -1 (S) : the partition of the plane formed by the farthest point Voronoi regions, their edges, and vertices

7. Farthest point Voronoi regions Construction of V -1 (7) Property The farthest point Voronoi regions are convex

8. Farthest point Voronoi regions Property If the farthest point Voronoi region of s is non empty then s is a vertex of conv(S)

9. Farthest point Voronoi regions Property If s is a vertex of conv(S) then the farthest point Voronoi region of s is non empty Property The farthest point Voronoi regions are unbounded

10. Farthest point Voronoi regions Property If s is a vertex of conv(S) then the farthest point Voronoi region of s is non empty Property The farthest point Voronoi regions are unbounded Corollary The farthest point Voronoi edges and vertices form a tree

11. Farthest point Voronoi edges and vertices edge : set of points equidistant from 2 sites and closer to all the others vertex : point equidistant from at least 3 sites and closer to all the others => c -1 (s,t) => u -1 (Q) x x x

12. Farthest point Voronoi diagram Question Which are the regions on both sides of c -1 (s,t) ? => V -1 (s) and V -1 (t) Question Which are the regions incident to u -1 (Q) ? V -1 (7) V -1 (2) V -1 (4) => V -1 (q) with q  Q x

13. Duality in planar partitions (reminder) Delaunay diagram : a partition dual to the Voronoi diagram => denoted by Del(S)

14. Farthest point Delaunay diagram Property The farthest point Delaunay edges do not intersect c -1 (2,4)

15. Farthest point Delaunay regions Property The farthest point Delaunay diagram of S is a partition of conv(S) in polygons inscribable in “full” circles u -1 (2,4,7) => Construction in O(n log n)

16. Order-2 Voronoi diagram V(s,t) : the set of points of the plane closer to each of s and t than to any other site Property The order-2 Voronoi regions are convex

17. Order-2 Voronoi regions V(s,t) : the set of points of the plane closer to each of s and t than to any other site Construction of V(3,5) Property The order-2 Voronoi regions are convex

18. Order-2 Voronoi edges and vertices edge : set of centers of circles passing through 2 sites and containing 1 site vertex : center of a circle passing through at least 3 sites and containing either 1 or 0 site => c p (s,t) => u p (Q) c 3 (1,2) u 5 (2,3,7) or u  (Q) u  (3,6,7,5)

19. Adjacency relations Question Which are the regions on both sides of c p (s,t) ? => V(p,s) and V(p,t) c 3 (1,2) V(2,3) V(1,3)

20. Adjacency relations Question Which are the regions on both sides of c p (s,t) ? => V(p,s) and V(p,t) Question Which are the regions incident to u p (Q) ? => V(p,q) with q  Q V(5,7) V(5,2) V(5,3)

21. Adjacency relations Question Which are the regions on both sides of c p (s,t) ? => V(p,s) and V(p,t) Question Which are the regions incident to u p (Q) ? => V(p,q) with q  Q Question Which are the regions incident to u  (Q) ? V(3,5) V(6,7) => V(q,q’) with q and q’ consecutive on the circle circumscribed to Q V(5,7) V(3,6)

22. Construction of Vor 2 (S) from Vor(S)

23. V(3) in Vor(S) The set of points of the plane closer to 3 than to any other site of S V(5) in Vor(S\{3}) The set of points of the plane closer to 5 than to any other site of S\{3}  V(3,5) => The intersection of V(3) and Vor(S\{3}) is a subset of Vor 2 (S)

24. Construction of Vor 2 (S) from Vor(S) V(5) in Vor(S) The set of points of the plane closer to 5 than to any other site of S V(3) in Vor(S\{5}) The set of points of the plane closer to 3 than to any other site of S\{5} => The intersection of V(5) and Vor(S\{5}) is a second subset of Vor 2 (S)  V(3,5)

25. Construction of Vor 2 (S) from Vor(S)  V(3,5)

26. Construction of Vor 2 (S) from Vor(S)

27. The algorithm For every region V(s) of Vor(S) - construct Vor(S\{s}) - compute the intersection of Vor(S\{s}) with V(s) Stick the different pieces together The time complexity Construction of one Voronoi diagram of n sites => O(n log n) Construction of n such diagrams => O(n 2 log n)

28. Construction of Vor 2 (S) from Vor(S)

29. Vor(S) and Vor 2 (S) V(3,5) c(3,5) V(6,7) c(6,7) c(1,3) V(1,3)

30. Vor(S) and Vor 2 (S) Property Every order-1 Voronoi edge c(s,t) is included in one and only one order-2 Voronoi region V(s,t). Conversely, every order-2 Voronoi region contains one and only one order-1 Voronoi edge. => c(s,t)  V(s,t)

31. Vor(S) and Vor 2 (S) Property Every order-1 Voronoi edge c(s,t) is included in one and only one order-2 Voronoi region V(s,t). Conversely, every order-2 Voronoi region contains one and only one order-1 Voronoi edge. x x  c(s,t) y y  c(s,t)

32. Vor(S) and Vor2(S) Corollary The order-1 and order-2 Voronoi edges do not intersect Corollary V(s,t) is an order-2 region iff c(s,t) is an order-1 edge V(s,t) is a region in Vor 2 (S) iff V(s) and V(t) share an edge in Vor(S)

33. Construction of Vor 2 (S) from Vor(S) A better algorithm For every region V(s) of Vor(S) - let S’ be the set of sites whose regions in Vor(S) share an edge with V(s) - construct Vor(S’) - compute the intersection of Vor(S’) with V(s) Stick the different pieces together Space complexity Number of regions of Vor 2 (S) = number of edges of Vor(S) => O(n)

34. Construction of Vor 2 (S) from Vor(S) Time complexity n s = number of edges of V(s) construction of 1 Vor(S’) => O(n s log n s ) construction of all Vor(S’) => O( (n s log n s )) < O( n s log n) < O(n log n)

35. Construction of Vor 2 (S) from Vor(S) Theorem The ordre-2 Voronoi diagram of n sites can be computed from the order-1 Voronoi diagram in O(n) time.

36. Order-2 Delaunay diagram Dual vertices The dual of an order-2 Voronoi region V(s,t) is the midpoint of st. => g(s,t)

37. Order-2 Delaunay vertices Property The order-2 Delaunay vertices are distinct points Proof g(s,t) a vertex of Del 2 (S) V(s,t) a region of Vor 2 (S) c(s,t) an edge of Vor(S) st an edge of Del(S)

38. Order-2 Delaunay edges V(4,1) V(4,6) g(4,1)g(4,6) c 4 (1,6) e 4 (1,6) Dual edges The dual of an order-2 Voronoi edge c p (s,t) is the segment connecting g(p,s) and g(p,t). => e p (s,t) e p (s,t) : image of st by an homothety of center p and ratio 1/2

39. Order-2 Delaunay edges Property The ordre-2 Delaunay edges do not intersect. Proof Let e p (s,t) and e p’ (s’,t’) be 2 distinct order-2 Delaunay edges.

40. Order-2 Delaunay edges Property The ordre-2 Delaunay edges do not intersect. Exercise Deal with the other cases. 

41. Order-2 Delaunay regions Region dual of u p (Q) Region dual of u  (Q) The polygon with vertices g(p,q) with q  Q. The polygon with vertices g(q,q’) with {q,q’} consecutive on the circle circumscribed to Q.

42. Constructing Del 2 (S) from Del(S) For every region of Del(S) compute its inscribed polygon For every site s of S - determine the polygon of the neighbors of s in Del(S) - construct the order-1 Delaunay diagram of this polygon - compute its image by an homothety with center s and ratio 1/2 Stick the different pieces together

43. Order-k Voronoi and Delaunay diagrams V(T) : the set of points of the plane closer to each site of T than to any other site (with |T| = k) g(T) : the center of gravity of T

44. Order-k Voronoi and Delaunay diagrams Theorem The size of the order-k diagrams is O(k(n-k)) Theorem The order-k diagrams can be constructed from the order-(k-1) diagrams in O(k(n-k)) time Corollary The order-k diagrams can be iteratively constructed in O(n log n + k 2 (n-k)) time