1. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) IP1: ADVANCED VORONOI AND DELAUNAY STRUCTURES Franz Aurenhammer IGI TU Graz Austria

2. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 1 PLANNED TOPICS (1) Shape Delaunay structures (1), (3), and (4) are related; start work there. (2) is of a different flavor, and maybe more tough. (2) Zone diagrams (3) Straight skeletons (4) Generalized medial axes

3. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 2 SHAPE DELAUNAY STRUCTURES Convex shape C (with center o), instead of empty circle. Edge inclusion, empty shape property. Voronoi diagram for convex distance function, take the dual (gives shape Delaunay for reflected shape)  Diagram does not change combinatorially, when center moves within C

4. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 3 SHAPE DELAUNAY STRUCTURES Not a full triangulation of the convex hull if C is not smooth  Support hull Direction-sensitivity to shape O(n logn) algorithms exist (D & C) Flipping works, too (criterion, termination)

5. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 4 SHAPE DELAUNAY STRUCTURES Flipping: Exclude certain edges Flipping criterion : Angles don‘t work any more Radii (Scaling factors of C): max(r,s) < max(t,u) can be shown min(r,s) < min(t,u) not true, i.g. If I contains points  no empty shape for edge exists, excluded. No point in II can give triangles with edge, as they cannot be covered by shape.

6. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 5 SHAPE DELAUNAY STRUCTURES Minimum spanning tree? d(C) is not a metric, unless C is symmetric d(C) depends on choice of center MST(C*) is in DT(C*) and the same as the MST w.r.t. m(C). The latter tree is part of DT(C) (as C* is union of shapes C)  Define symmetric metric m(C) = d(C*) (C* = Minkowski sum of C and C‘)

7. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 6 SHAPE DELAUNAY STRUCTURES Open questions: -- More optimization properties (Flip: r+s < t+u ?) -- Number of flips, given C ? -- Deciding whether T is some shape Delaunay (find C) Characterization of DT(C)... -- Is the number of shape Delaunays polynomial for every point set?

8. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 7 ZONE DIAGRAMS Quite new, challenging concept Set S of n points in the plane. Zone of p in S: Domain closer to p than to any other zone [Asano et al.]  Implicit definition, neutral zone, existence, uniqueness? `Kingdom interpretation´

9. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 8 ZONE DIAGRAMS Some properties Zones are convex, and contained in Voronoi region. Zones can expand, when site is added. Trisector of two sites, algebraic? ∩ with line...

10. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 9 ZONE DIAGRAMS Approximation Family of subsets A = (A 1,...,A n ), consider B = (B 1,...,B n ) with B i = dom(p i, UA j ) Operator OP, B = OP(A) Start with family Z 0 = (p 1,...,p n ) Iterate Z i+1 = OP(Z i ) Z 1 = (reg(p 1 ),...,reg(p n )) Even (odd) Z i gives inner (outer) approximation of the zone diagram Fixed point of OP is the zone diagram of p 1,...,p n

11. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 10 ZONE DIAGRAMS `Mollified version´: Territory diagrams Zone diagram: Z(p i ) = dom(p i, UZ(p j ) Territory diagram: T(p i ) contained in dom(p i, UT(p j ) (Equality is replaced by inclusion) Trivial cases... Territories can be larger than zones, and even nonconvex Maximal territory diagrams (cannot be expanded) Zone diagrams are not the only instance...

12. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 11 ZONE DIAGRAMS Polynomial root finding Set of n points in the plane...interpreted as the roots of a complex polynomial of degree n Root finding: Initial value z 0, iterative method. Convergence to some root z i = point location, yielding site p i  Basin of attraction for each site Seqence of iteration functions B m [Kalantari] B 2 = Newton‘s method, B 3 = Halley‘s method Fractal behavior... Uniform approximation of Voronoi diagram, as m grows.

13. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 12 ZONE DIAGRAMS Similarity to zone diagram `Save´ convex areas inside immediate basins of attraction

14. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 13 ZONE DIAGRAMS Open questions More insight into the structure of zone diagrams Construction algorithms, approximation... Is there a link to basins of attraction?

15. EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI) 14 Thank you