Higher-Order Delaunay Triangulations

1. Higher-Order Delaunay Triangulations Marc van Kreveld Presentation based on joint work with: Joachim Gudmundsson, Mikael Hammar, Herman Haverkort, Thierry de Kok, Maarten Löffler, Rodrigo Silveira

2. Overview • Motivation • Triangulation for terrains • Higher order Delaunay triangulations • Basics • First order Delaunay triangulation results • Minimizing local minima in terrains • Higher order triangulations of polygons

3. Polyhedral terrains, or TINs • Points with (x,y) and elevation as input • TIN as terrain representation • Choice of triangulation is important 25 29 25 29 24 24 21 21 19 19 78 78 73 73 15 15 10 10 12 12

4. Realistic terrains • Due to erosion, realistic terrains • have few local minima • have valley lines that continue local minimum, interrupted valley line after an edge flip

5. Terrain modeling in GIS • Terrain modeling is extensively studied in geomorphology and GIS • Need to avoid artifacts like local minima • Need correct “shape” for run-off models, hydrological models, avalanche models, ... 17 52 6 12 local minimum in a TIN 21 24

6. Terrain modeling in GIS • Terrain convexity/concavity in cross-sections also influencessurface flow interest inplan curvatureandprofile curvature

7. Delaunay triangulation • Maximizes minimum angle • Empty circle property

8. Delaunay triangulation • Does not take elevation into account • May give local minima • May give interrupted valleys • Does not consider curvature

9. Triangulate to minimize local minima?

10. Triangulate to minimize local minima? Connect everything to global minimum and complete bad triangle shape & interpolation

11. Higher order Delaunay triangulations • Compromise between good shape & interpolation, and flexibility (w.r.t. DT) to satisfy other constraints • k -th order: allow k points in circle 1st order 4th order 0th order

12. Higher order Delaunay triangulations • Introduced by Gudmundsson, Hammar, and van Kreveld (ESA 2000, CGTA 2004) • Delaunay triangulation = 0-th order DT • A triangulation is k-th orderDelaunay if the circumcircleof each of its triangles has≤ k points inside

13. Higher order Delaunay triangulations • All edges that may be in an order-4 Delaunay triangulation

14. Higher order Delaunay triangulations • uv is an order-k Delaunay edgethe order-(k+1) VD has cells for{u, p1,..., pk} and {v, p1,..., pk} (= the bisector of uv exists in the order-(k+1) VD {u, p1, p2,p3} p1 p3 v u p2 {v, p1, p2,p3}

15. Higher order Delaunay triangulations • Useful order-k Delaunay edges: edges that can be used in an order-k DT useful order 5 Delaunay edge

16. Higher order Delaunay triangulations • Computing all useful order-k Delaunay edges takes O(nk log n + nk2) time: • Compute the order-(k+1) VD to get order-k edges • Test each edge in O(k + log n) time for usefulness • Trace the edge in the DT • Determine the two circles • Query with them:≤ k points inside? find kth closestpoint from center

17. Open: Given n edges, complete them to the lowest order DT (solved if all edges have useful order at most 3) Higher order Delaunay triangulations • A useful order-k Delaunay edge can be used in an order-k DT, just take the CDT with the edge • But: two (or more) useful order-k Delaunay triangulations may give an order-(2k−2) DT • The CDT guarantees order 2k−2 • Sometimes the CDT gives order 2k−2 but another triangulation gives order k

18. Higher order Delaunay triangulations • Gudmundsson, Hammar, vK (2000) • higher order Delaunay triangulations • Gudmundsson, Haverkort, vK (2003) • constrained higher order Delaunay triangulations • de Kok, vK, Löffler (2005) • local minima, NP-hardness, drainage, experiments • vK, Löffler, Silveira (2006/2007) • first order DT, polynomial, NP-hardness, approximation • Silveira, vK (2007) • polygons, dynamic programming, FPT, experiments

19. Open: - Given n points and k 2 (constant), is minimizing local minima over all order-k DT NP-hard? - Is there an approximation with a factor better than O(k2)? Minimizing local minima • Minimizing local minima for order-k DT is NP-hard if k = (n) We study two heuristics (flip and hull) for reducing local minima on terrains, and one (valley) making contiguous drainage networks

20. Experiments on terrains

21. Quinn Peak • Elevation grid of 382 x 468 • Random sample of 1800 vertices • Delaunay triangulation • 53 local minima

22. Hull heuristic applied • Order 4 Delaunay triangulation • 25 local minima

23. Hull heuristic Flip heuristic

24. Delaunay triangulation

25. Hull-8 + valley heuristic

26. Experimental results • Hull and Flip reduce local minima by 60−70% for order 8; Hull is often better • Hull and Flip are near-optimal for orders up to 8 • Valley reduces the number of valley edge components by 20−40% for order 8 • Hull + Valley seems best

27. First order Delaunay triangulations • First order Delaunay triangulations have a simple structure • all certain edges (Delaunay) give a subdivision in triangles and quadrilaterals • all possible edgesare diagonals ofthe quadrilaterals

28. First order Delaunay triangulations • Minimizing local minima is easy: choose the diagonal that connects to the lowest point ofthe quadrilateral O(n log n) timefor any n-point set 12 8 4 7 9 5 2

29. First order Delaunay triangulations • Also simple: measures that relate to individual edges or triangles (or is composed of it), like • min max triangle area • min max angle • min total edge length • min sum of inscribedcircle radii • ...

30. First order Delaunay triangulations • Not trivial • min max vertex degree • min max area ratio (across edges) • min max spatial angle(across edges) • max no. of convex edges • min no. of mixed vertices plane terrain A vertex v is mixed if it does not lie onthe 3d convex hull of {v} neighbors(v)= a plane through v exists with all neighbors to one side

31. NP-hard to decide if degree ≤ 20 can be achieved; reduction from planar 3-SAT NP-hard; reduction from planarMAX-2-SAT NP-hard; reduction from planar 3-SAT First order Delaunay triangulations • Not trivial: NP-hard • min max vertex degree • min max area ratio(across edges) • min max spatial angle(across edges) • max no. of convex edges • min no. of mixed vertices

32. (1−)-approx in 2O(1/)·n time First order Delaunay triangulations • Not trivial: approximation • min max vertex degree • min max area ratio(across edges) • min max spatial angle(across edges) • max no. of convex edges • max no. of non-mixed vertices No PTAS possible; 3/2-approx exists (1−)-approx in 2O(1/ ) ·n time 2

33. First order Delaunay triangulations • Not trivial: polynomial • min max area ratio (across edges) • min max spatial angle (across edges) • Still O(n log n) time: • Sort the O(n) candidate values • Do a binary search; each decisioninvolves building an O(n) size2-SAT formula where the diagonalrepresents true or false xi = true xi = false xj = true xj = false (xixj)

34. Higher order DT for polygons • Can we optimally triangulate a polygon P over all order-k DTs (min max area; min weight; ...) ? Extension 1: there may be points outside P that influence the order of triangles in P Extension 2: there may be points or components inside P

35. Higher order DT for polygons • Optimal triangulation of a polygon by dynamic programming for decomposable measures, typically in O(n3) time (Klincsek, 1980) w u OPT(u,v) = max / min v w between u,v { OPT(u,w)  OPT(v,w) }

36. Higher order DT for polygons • For order-k DT: • First determine all order-k Delaunay triangles in P • Only use these in the dynamic program The O(nk2) order-k Delaunay triangles can be determined in O(nk2 log k + kn log n) time (also for extension 1) w u v Dynamic programming: O(nk) optimal subproblems, O(k) choices  O(nk2) time

37. Higher order DT for polygons • When components are inside: • Connect topmost point of each component to get one polygon, in all possible ways • Apply the DP algorithm for each polygon For h components there are O(nh) connections, but for each component we can restrict ourselves to O(k) connections The whole algorithm takes O(nk log n + kh+2n) time, so FPT for constant k

38. Higher order DT for polygons • Why only O(k) connections? • The topmost point t must have a Delaunay edge tu up • Any Delaunay edge intersects O(k) order-k Delaunay edges (GHK 2000) • The lowest one, vw, used in OPT gives that vt and wt are also in OPT; at least one of them is upward u So for one upward Delaunay edge tu from t, only try tu and all upper endpoints of the order-k Delaunay edges that intersect tu w t v

39. Higher order DT for polygons • For a point set, if the order is low, there are many fixed edges and few components Order 4; blue edgesare in every order-4 DT

40. (2O(k)/2) 2 Conclusions and future work • Theory: • NP-hardness of minimizing local minima for small k ? • Completion of edges to lowest order DT ? • The PTAS for max no. of convex edges for order-1 DT extends to order k,but is doubly-exponential in k: • The PTAS for max no. of non-mixed vertices does not seem to extend • Practice: • Up to what order can terrain criteria be solved optimally in reasonable time? • Can flow processes be modelled well enough?

41. Based on ... • Gudmundsson, Hammar, vK (2000)Higher order DTs • Gudmundsson, Haverkort, vK (2003)Constrained higher order DTs • de Kok, vK, Löffler (2005) Generating realistic terrains with higher-order DTs • vK, Löffler, Silveira (2006/2007)Optimization for first order DTs • Silveira, vK (2007)Optimal higher order DTs of polygons