Triangulation

1
Triangulation

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• ????????
• deng_at_tsinghua.edu.cn
• http//vis.cs.tsinghua.edu.cn10020/deng
• 2016?10?19??????7?1?

2
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(nlogn) 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
  /

3
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

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Greedy triangulation

• Idea
• a simulation of the Kruskal algorithm for MST
• Optimal triangulation
• not guaranteed
• optimal triangulation ?
• later

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Greedy Triangulation

• Algorithms
• Naive O(n3) O(n2)
• Gi79 O(n2logn) O(n2)
• Go89Li88 O(n2logn) O(n)
• LL92Wa93 O(n2) O(n)
• Lev92Wa94 O(nlogn) O(n) (without necessary
  proofs)
• LK99 O(nlogn) O(n) (with proof)
• Special cases
• Delaunay Triangulation ---O(n)--gt GT
• EMST ---O(n)--gt 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

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Minimum Weight Triangulation

• Research directions
• hardness of MWT
• MWT algorithms
• compute MWT subgraphs
• Greedy triangulation
• approximates the MWT quite well
• expected-O(n) algorithm

9
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