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Triangulation 邓俊辉 清华大学计算机系 deng@tsinghua. edu. cn http: //vis. cs. tsinghua. edu. cn: 10020/~deng 3/19/201811: Triangulation 邓俊辉 清华大学计算机系 [email protected] edu. cn http: //vis. cs. tsinghua. edu. cn: 10020/~deng 3/19/201811: 07 PM

Triangulation of Polygons ☞ Polygon – Existence: – Uniqueness: yes with/without holes no usually Triangulation of Polygons ☞ Polygon – Existence: – Uniqueness: yes with/without holes 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 / … Junhui Deng, Tsinghua Computer 2

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

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

Greedy Triangulation ☞ Algorithms – Naive O(n 3) | O(n 2) – [Gi 79] Greedy Triangulation ☞ Algorithms – Naive O(n 3) | O(n 2) – [Gi 79] O(n 2 logn) | O(n 2) – [Go 89][Li 88] O(n 2 logn) | O(n) – [LL 92][Wa 93] O(n 2) | O(n) – [Lev 92][Wa 94] O(nlogn) | O(n) (without necessary proofs) – [LK 99] O(nlogn) | O(n) (with proof) ☞ Special cases – Delaunay Triangulation ---O(n)--> GT – EMST ---O(n)--> DT Junhui Deng, Tsinghua Computer 5

Delaunay triangulation ☞ O(nlogn) ☞ Parabolic transformation – Convex hull / convex polytope – 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 Junhui Deng, Tsinghua Computer 6

Minimum Weight Triangulation ☞ MWT – min {edge lengths} – best criterion for optimal Minimum Weight Triangulation ☞ MWT – min {edge lengths} – best criterion for optimal triangulation – == Delaunay triangulation? ☞ P? NP-Complete? – open problem list of Garey and Johnson [GJ 79] – Several published "algorithms" have been found to be incorrect ☞ Generalized version with non-Euclidean distances – NP-complete E. L. Lloyd, 1977 Junhui Deng, Tsinghua Computer 7

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

Steiner Triangulation ☞ Steiner points – vertex set U is required to be a 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 l all angles are bounded between 36 o and 80 o, – in O(poly(card(U)) time, – using O(card(V)) Steiner points Junhui Deng, Tsinghua Computer 9