[PPT] - Flips in higher order Delaunay triangulations Elena Arseneva, PowerPoint Presentation

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Flips in higher order Delaunay triangulations

Elena Arseneva, Prosenjit Bose, Pilar Cano, and Rodrigo I. Silveira

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Delaunay triangulations

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Delaunay triangulations

S

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Delaunay triangulations

DT(S)

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Delaunay triangulations

DT(S)

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Delaunay triangulations

DT(S)

The DT(S) is unique

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Delaunay triangulations

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Delaunay triangulations

Dual of the Voronoi diagram

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Delaunay triangulations

Dual of the Voronoi diagram
Contains the minimum spanning tree of a point set

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Delaunay triangulations

Dual of the Voronoi diagram
Contains the minimum spanning tree of a point set
Maximizes the minimum angle

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Delaunay triangulations

Dual of the Voronoi diagram
Contains the minimum spanning tree of a point set
Maximizes the minimum angle

Many applications in Graphics, GIS, mesh generation, among others

“well-shaped” triangles

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Higher order Delaunay triangulations

Yet...

Picture from: Gudmundsson, J., Hammar, M., van Kreveld, M.: Higher order Delaunay triangulations. Comput.

Geom. 23(1), 8598 (2002)

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Higher order Delaunay triangulations

Yet...

Picture from: Gudmundsson, J., Hammar, M., van Kreveld, M.: Higher order Delaunay triangulations. Comput.

Geom. 23(1), 8598 (2002)

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Higher order Delaunay triangulations

S

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Higher order Delaunay triangulations

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Higher order Delaunay triangulations

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Higher order Delaunay triangulations

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Higher order Delaunay triangulations

Also called order-k triangulations

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Higher order Delaunay triangulations

=

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The flip graph

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The flip graph

Flip

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The flip graph

Flip

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The flip graph

Flip graph

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The flip graph and the Delaunay triangulation

Illegal edge

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The flip graph and the Delaunay triangulation

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The flip graph and the Delaunay triangulation

Any triangulation can be transformed into the DT

by flipping only illegal edges.

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The flip graph and the Delaunay triangulation

Any triangulation can be transformed into the DT

by flipping only illegal edges.

The flip graph of any point set S, denoted T (S), is

connected.

The distance in T (S) between any two

triangulations of S is O(n2).

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The flip graph and order-k triangulations

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The flip graph and order-k triangulations

Abe and Okamoto observed that the flip graph of
rder-k triangulations of a point set S, denoted

Tk(S), is connected for k ≤ 2.

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The flip graph and order-k triangulations

Abe and Okamoto observed that the flip graph of
rder-k triangulations of a point set S, denoted

Tk(S), is connected for k ≤ 2.

rder-3 edge

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The flip graph and order-k triangulations

Abe and Okamoto observed that the flip graph of
rder-k triangulations of a point set S, denoted

Tk(S), is connected for k ≤ 2.

rder-3 edge
rder-k triangulation
rder-k edges

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The flip graph and order-k triangulations

Abe and Okamoto observed that the flip graph of
rder-k triangulations of a point set S, denoted

Tk(S), is connected for k ≤ 2.

rder-3 edge
rder-k triangulation
rder-k edges

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The flip graph and order-k triangulations

Abe and Okamoto observed that the flip graph of
rder-k triangulations of a point set S, denoted

Tk(S), is connected for k ≤ 2. About the flip graph of triangulations with order-k edges, Abellans et al. showed:

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The flip graph and order-k triangulations

Abe and Okamoto observed that the flip graph of
rder-k triangulations of a point set S, denoted

Tk(S), is connected for k ≤ 2. About the flip graph of triangulations with order-k edges, Abellans et al. showed:

Is connected when k ≤ 1
Might be disconnected when k ≥ 2
For any k, the flip graph is connected when S is in

convex position

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The flip graph and order-k triangulations

Abe and Okamoto observed that the flip graph of
rder-k triangulations of a point set S, denoted

Tk(S), is connected for k ≤ 2. About the flip graph of triangulations with order-k edges, Abellans et al. showed:

Is connected when k ≤ 1
Might be disconnected when k ≥ 2
For any k, the flip graph is connected when S is in

convex position exponential upper bound

n the flip distance

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Our results

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Our results

For any k there exists a point set S in convex position where

Tk(S) is disconnected. Moreover, k − 1 flips are sometimes necessary in order to transform an order-k triangulation of S into another.

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Our results

For any k there exists a point set S in convex position where

Tk(S) is disconnected. Moreover, k − 1 flips are sometimes necessary in order to transform an order-k triangulation of S into another.

For any order-k triangulation of a point set in convex position

there exists an order-k triangulation at distance at most k − 1 in T2k−2(S)

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Our results

For any k there exists a point set S in convex position where

Tk(S) is disconnected. Moreover, k − 1 flips are sometimes necessary in order to transform an order-k triangulation of S into another.

For any order-k triangulation of a point set in convex position

there exists an order-k triangulation at distance at most k − 1 in T2k−2(S)

Let k = 2, 3, 4, 5. For any order-k triangulation of a point set

in general position there exists an order-k triangulation at distance at most k − 1 in T2k−2(S)

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Our results

For any k there exists a point set S in convex position where

Tk(S) is disconnected. Moreover, k − 1 flips are sometimes necessary in order to transform an order-k triangulation of S into another.

For any order-k triangulation of a point set in convex position

there exists an order-k triangulation at distance at most k − 1 in T2k−2(S)

Let k = 2, 3, 4, 5. For any order-k triangulation of a point set

in general position there exists an order-k triangulation at distance at most k − 1 in T2k−2(S)

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A lower bound

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A lower bound

u v

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A lower bound

u v p1

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A lower bound

u v p1 q1

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A lower bound

u v p1 q1

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A lower bound

u v p1 q1

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A lower bound

u v p1 q1 p2 p3 q2 q3

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A lower bound

u v p1 q1 p2 p3 q2 q3

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A lower bound

u v p1 q1 p2 p3 q2 q3

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A lower bound

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A lower bound

Theorem. For any k > 2 there is a set Sk of 2k + 2 points in

convex position such that G(Tk(Sk)) is not connected. Moreover, there is a triangulation Tk in Tk(Sk) such that in order to transform Tk into any other triangulation in Tk(Sk) one needs to perform at least k − 1 flips.

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An upper bound for convex position

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An upper bound for convex position

Theorem. For a point set S in convex position and k ≥ 2, let

T = DT(S) be a triangulation in Tk(S). Then, there exists T ′ in Tk(S) such that there is a path between T and T ′ in G(T2k−2(S))

f length at most k − 1, where each edge of the path corresponds

to flipping an illegal edge.

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An upper bound for convex position

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An upper bound for convex position

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An upper bound for convex position

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An upper bound for convex position

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An upper bound for convex position

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An upper bound for convex position

Theorem. For a point set S in convex position and k ≥ 2, let

T = DT(S) be a triangulation in Tk(S). Then, there exists T ′ in Tk(S) such that there is a path between T and T ′ in G(T2k−2(S))

f length at most k − 1, where each edge of the path corresponds

to flipping an illegal edge.

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Summary

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We showed that k − 1 flips might be necessary for transforming
ne k-order triangulation to any other k-order triangulation
For k ≥ 2, we showed that any order-k triangulation can be

transformed into some other order-k triangulation by at most k − 1 flips of only illegal edges, such that the intermediate triangulations are of order 2k − 2, in the following settings: a) For any k ≥ 2 and points in convex position b) For k = 2, 3, 4, 5 and points in general position

Summary

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We showed that k − 1 flips might be necessary for transforming
ne k-order triangulation to any other k-order triangulation
For k ≥ 2, we showed that any order-k triangulation can be

transformed into some other order-k triangulation by at most k − 1 flips of only illegal edges, such that the intermediate triangulations are of order 2k − 2, in the following settings: a) For any k ≥ 2 and points in convex position b) For k = 2, 3, 4, 5 and points in general position

Consequences

Summary

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We showed that k − 1 flips might be necessary for transforming
ne k-order triangulation to any other k-order triangulation
For k ≥ 2, we showed that any order-k triangulation can be

transformed into some other order-k triangulation by at most k − 1 flips of only illegal edges, such that the intermediate triangulations are of order 2k − 2, in the following settings: a) For any k ≥ 2 and points in convex position b) For k = 2, 3, 4, 5 and points in general position

Consequences

Any order-k triangulation can be transformed into another by

O(kn) flips

An efficient enumeration algorithm of order-k triangulations

Summary

SLIDE 64

We showed that k − 1 flips might be necessary for transforming
ne k-order triangulation to any other k-order triangulation
For k ≥ 2, we showed that any order-k triangulation can be

transformed into some other order-k triangulation by at most k − 1 flips of only illegal edges, such that the intermediate triangulations are of order 2k − 2, in the following settings: a) For any k ≥ 2 and points in convex position b) For k = 2, 3, 4, 5 and points in general position

Consequences

Any order-k triangulation can be transformed into another by

O(kn) flips

An efficient enumeration algorithm of order-k triangulations