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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

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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

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Polyhedral terrains, or TINs Points with (x, y) and elevation as input TIN as terrain representation Choice of triangulation is important

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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

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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, local minimum in a TIN

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Terrain modeling in GIS Terrain convexity/concavity in cross-sections also influences surface flow interest in plan curvature and profile curvature

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Delaunay triangulation Maximizes minimum angle Empty circle property

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

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Triangulate to minimize local minima?

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Connect everything to global minimum and complete bad triangle shape & interpolation

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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 1 st order 0 th order 4 th order

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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 order Delaunay if the circumcircle of each of its triangles has k points inside

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

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Higher order Delaunay triangulations uv is an order-k Delaunay edge the order-(k +1) VD has cells for {u, p 1,..., p k } and {v, p 1,..., p k } (= the bisector of uv exists in the order-(k +1) VD u v p1p1 p2p2 {u, p 1, p 2, p 3 } {v, p 1, p 2, p 3 } p3p3

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

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Higher order Delaunay triangulations Computing all useful order-k Delaunay edges takes O(nk log n + nk 2 ) 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 k th closest point from center

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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-(2k2) DT –The CDT guarantees order 2k2 –Sometimes the CDT gives order 2k2 but another triangulation gives order k Open: Given n edges, complete them to the lowest order DT (solved if all edges have useful order at most 3)

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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

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Minimizing local minima Minimizing local minima for order-k DT is NP-hard if k = (n ) 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(k 2 )? We study two heuristics (flip and hull) for reducing local minima on terrains, and one (valley) making contiguous drainage networks

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Experiments on terrains

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

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Hull heuristic applied Order 4 Delaunay triangulation 25 local minima

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Hull heuristic Flip heuristic

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Delaunay triangulation

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Hull-8 + valley heuristic

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Experimental results Hull and Flip reduce local minima by 6070% 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 2040% for order 8 Hull + Valley seems best

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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 edges are diagonals of the quadrilaterals

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First order Delaunay triangulations Minimizing local minima is easy: choose the diagonal that connects to the lowest point of the quadrilateral O(n log n) time for any n-point set

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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 inscribed circle radii –...

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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 A vertex v is mixed if it does not lie on the 3d convex hull of {v} neighbors(v) = a plane through v exists with all neighbors to one side terrain plane

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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 NP-hard to decide if degree 20 can be achieved; reduction from planar 3-SAT NP-hard; reduction from planar 3-SAT NP-hard; reduction from planar MAX-2-SAT

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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 2 O(1/ ) ·n time (1 )-approx in 2 O(1/ ) ·n time 2

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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 decision involves building an O(n) size 2-SAT formula where the diagonal represents true or false x i = true x i = false x j = true x j = false (x i x j )

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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

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Higher order DT for polygons Optimal triangulation of a polygon by dynamic programming for decomposable measures, typically in O(n 3 ) time (Klincsek, 1980) u v w OPT(u,v) = max / min { OPT(u,w) OPT(v,w) } w between u,v

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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 u v w The O(nk 2 ) order-k Delaunay triangles can be determined in O(nk 2 log k + kn log n) time (also for extension 1) Dynamic programming: O(nk) optimal subproblems, O(k) choices O(nk 2 ) time

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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(n h ) connections, but for each component we can restrict ourselves to O(k) connections The whole algorithm takes O(nk log n + k h+2 n) time, so FPT for constant k

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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 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 u w t v

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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 edges are in every order-4 DT

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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? 2 (2 O(k) / 2 )

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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

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