Triangulation

邓俊辉

清华大学计算机系deng@tsinghua.edu.cn

http://vis.cs.tsinghua.edu.cn:10020/~deng

2023年4月18日下午 07:42

22Junhui 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 / …

33Junhui 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

– …

44Junhui Deng, Tsinghua Computer

Greedy triangulation

☞ Idea

– a simulation of the Kruskal algorithm for MST

☞ Optimal triangulation

– not guaranteed

– optimal triangulation = ? later …

55Junhui Deng, Tsinghua Computer

Greedy Triangulation

☞ Algorithms

– Naive O(n3) | O(n2)

– [Gi79] O(n2logn) | O(n2)

– [Go89][Li88] O(n2logn) | O(n)

– [LL92][Wa93] O(n2) | 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

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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

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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-complete E. L. Lloyd, 1977

88Junhui 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

99Junhui 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 36o and 80o,

– in O(poly(card(U)) time,

– using O(card(V)) Steiner points