2. Triangulation 邓俊辉 清华大学计算机系 deng@tsinghua.edu.cn http://vis.cs.tsinghua.edu.cn:10020/~deng 2015年5月24日星期日 2015年5月24日星期日 2015年5月24日星期日 2015年5月24日星期日 2015年5月24日星期日 2015年5月24日星期日下午4时0分 下午4时0分 下午4时0分 下午4时0分 下午4时0分

3. 2 Junhui Deng, Tsinghua Computer Triangulation of Polygons ☞ Polygon –Existence:yes with/without holes –Uniqueness:no usually ☞ Simple polygons – O(nlogn)Garey et al., 1978 – O(nloglogn)R. E. Tarjan & C. J. Van Wyk, 1988 –expected-O(nlog * n)K. Clarkson, R. E. Tarjan and C. J. Van Wyk, 1989 – O(n)B. Chazelle, 1991 (by trapezoidalization) –expected-O(n)N. M. Amato, et al., 2001 (by randomization) ☞ Art gallery ☞ Optimal triangulation –e.g., minimum ink / maximizing the minimum angle / …

4. 3 Junhui Deng, Tsinghua Computer Triangulation of Point Sets ☞ Problem –Give a set V of n points in the plane construct a maximal planar graph T(V) taking V as the vertex set –Existence! –Uniqueness? ☞ Algorithms –Brute-force –Plane-sweep –…–…

5. 4 Junhui Deng, Tsinghua Computer Greedy triangulation ☞ Idea –a simulation of the Kruskal algorithm for MST ☞ Optimal triangulation –not guaranteed –optimal triangulation = ? later …

6. 5 Junhui Deng, Tsinghua Computer Greedy Triangulation ☞ Algorithms –NaiveO(n 3 ) | O(n 2 ) –[Gi79]O(n 2 logn) | O(n 2 ) –[Go89][Li88]O(n 2 logn) | O(n) –[LL92][Wa93]O(n 2 ) | O(n) –[Lev92][Wa94]O(nlogn) | O(n)(without necessary proofs) –[LK99]O(nlogn) | O(n)(with proof) ☞ Special cases –Delaunay Triangulation ---O(n)--> GT –EMST ---O(n)--> DT

7. 6 Junhui Deng, Tsinghua Computer Delaunay triangulation ☞ O(nlogn) ☞ Parabolic transformation –Convex hull / convex polytope –lower envelope ~ Delaunay triangulation –upper envelope ~ Delaunay farthest triangulation of convex polygon ☞ Nearest Neighbor Query –Given a set S of n sites, for each site find its nearest neighboring site –Notice: NN is NOT a mutual relationship –The Delaunay edges include all the nearest neighbor pairs ☞ EMST(S)  RNG(S)  GG(S)  DT(S) –Gabriel Graph –Relative Neighborhood Graph –Euclidean Minimum Spanning Tree

8. 7 Junhui Deng, Tsinghua Computer Minimum Weight Triangulation ☞ MWT – min  {edge lengths} –best criterion for optimal triangulation –== Delaunay triangulation? ☞ P? NP-Complete? – open problem list of Garey and Johnson [GJ79] –Several published "algorithms" have been found to be incorrect ☞ Generalized version with non-Euclidean distances –NP-completeE. L. Lloyd, 1977

9. 8 Junhui Deng, Tsinghua Computer Minimum Weight Triangulation ☞ Research directions –hardness of MWT –MWT algorithms –compute MWT subgraphs ☞ Greedy triangulation –approximates the MWT quite well –expected-O(n) algorithm

10. 9 Junhui Deng, Tsinghua Computer Steiner Triangulation ☞ Steiner points –vertex set U is required to be a superset of V –reasonable in FEA –it is required card(U) = O(card(V)) -- why? ☞ Algorithms based on quadtrees –constructs a Steiner triangulation, where all angles are bounded between 36 o and 80 o, –in O(poly(card(U)) time, –using O(card(V)) Steiner points