Making Triangulations 4-connected using Flips Prosenjit Bose, Dana - - PowerPoint PPT Presentation

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Making Triangulations 4-connected using Flips Prosenjit Bose, Dana - - PowerPoint PPT Presentation

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Making Triangulations 4-connected using Flips Prosenjit Bose, Dana Jansens, Andr e van Renssen, Maria Saumell and Sander Verdonschot Carleton University December 19, 2011 Sander Verdonschot (Carleton University) Making Triangulations

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SLIDE 1

Making Triangulations 4-connected using Flips

Prosenjit Bose, Dana Jansens, Andr´ e van Renssen, Maria Saumell and Sander Verdonschot

Carleton University

December 19, 2011

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 1 / 21

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SLIDE 2

Flips

Replace one diagonal of a quadrilateral with the other

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 2 / 21

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SLIDE 3

Flips

Replace one diagonal of a quadrilateral with the other

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 2 / 21

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SLIDE 4

Flips

Replace one diagonal of a quadrilateral with the other

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 2 / 21

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SLIDE 5

Flip Graph

Vertex for each triangulation Edge if two triangulations differ by one flip

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 3 / 21

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SLIDE 6

Flip Graph

Vertex for each triangulation Edge if two triangulations differ by one flip Flip Distance: shortest path in flip graph

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 3 / 21

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SLIDE 7

Flip Graph

Connected?

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 4 / 21

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SLIDE 8

Flip Graph

Connected?

Yes - Wagner (1936)

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 4 / 21

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SLIDE 9

Flip Graph

Connected?

Yes - Wagner (1936)

Diameter?

O(n2) - Wagner (1936)

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 4 / 21

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SLIDE 10

Flip Graph

Connected?

Yes - Wagner (1936)

Diameter?

O(n2) - Wagner (1936) 8n − 54 - Komuro (1997) 6n − 30 - Mori et al. (2003)

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 4 / 21

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SLIDE 11

Algorithm

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 5 / 21

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SLIDE 12

Algorithm

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 5 / 21

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SLIDE 13

Algorithm

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 5 / 21

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SLIDE 14

Algorithm Mori et al.

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 6 / 21

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SLIDE 15

Algorithm Mori et al.

4-connected ⇒ Hamiltonian

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 6 / 21

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SLIDE 16

Algorithm Mori et al.

4-connected ⇒ Hamiltonian n − 4 2n − 11 Total: 6n − 30

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 6 / 21

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SLIDE 17

Algorithm Mori et al.

4-connected ⇒ Hamiltonian n − 4 2n − 11 Total: 6n − 30

3n−6 5

5.2n − 24.4

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 6 / 21

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SLIDE 18

Making triangulations 4-connected

Separating triangle: 3-cycle whose removal disconnects the graph

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 7 / 21

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SLIDE 19

Making triangulations 4-connected

Separating triangle: 3-cycle whose removal disconnects the graph No separating triangles ⇐ ⇒ 4-connected

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 7 / 21

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SLIDE 20

Making triangulations 4-connected

Separating triangle: 3-cycle whose removal disconnects the graph No separating triangles ⇐ ⇒ 4-connected Flipping an edge of a separating triangle removes it

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 7 / 21

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SLIDE 21

Making triangulations 4-connected

Separating triangle: 3-cycle whose removal disconnects the graph No separating triangles ⇐ ⇒ 4-connected Flipping an edge of a separating triangle removes it Prefer shared edges

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 7 / 21

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SLIDE 22

Upper Bound

To prove: #flips ≤ (3n − 6)/5

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 8 / 21

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SLIDE 23

Upper Bound

To prove: #flips ≤ (3n − 6)/5 Charging scheme:

Coin on every edge Pay 5 coins per flip

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 8 / 21

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SLIDE 24

Paying for flips

Invariant: Every edge of a separating triangle has a coin Charge the flipped edge Charge all edges that aren’t shared

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 9 / 21

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SLIDE 25

Paying for flips

Free edge: edge that is not part of any separating triangle

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 10 / 21

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SLIDE 26

Paying for flips

Free edge: edge that is not part of any separating triangle Every vertex of a separating triangle is incident to a free edge inside the triangle

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 10 / 21

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SLIDE 27

Paying for flips

Free edge: edge that is not part of any separating triangle Invariant: Every vertex of a separating triangle is incident to a free edge inside the triangle that has a coin

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 10 / 21

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SLIDE 28

Paying for flips

Free edge: edge that is not part of any separating triangle Invariant: Every vertex of a separating triangle is incident to a free edge inside the triangle that has a coin Charge all free edges that aren’t needed by other separating triangles

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 10 / 21

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SLIDE 29

Which edges to flip?

A deepest separating triangle is contained in the maximum number of separating triangles

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 11 / 21

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SLIDE 30

Which edges to flip?

A deepest separating triangle is contained in the maximum number of separating triangles Flip:

An arbitrary edge Shared with other separating triangles Not shared with a containing triangle

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 11 / 21

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SLIDE 31

Which edges to flip?

A deepest separating triangle is contained in the maximum number of separating triangles Flip:

An arbitrary edge Shared with other separating triangles Not shared with a containing triangle

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 11 / 21

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SLIDE 32

Which edges to flip?

A deepest separating triangle is contained in the maximum number of separating triangles Flip:

An arbitrary edge Shared with other separating triangles Not shared with a containing triangle

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 11 / 21

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SLIDE 33

Which edges to flip?

Case 1: No shared edges We can charge: The flipped edge An unshared triangle edge An unshared free edge A superfluous free edge

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 12 / 21

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SLIDE 34

Which edges to flip?

Case 2: Shares edges with non-containing triangles We can charge: The flipped edge An unshared triangle edge An unshared free edge A superfluous free edge

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 13 / 21

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SLIDE 35

Which edges to flip?

Case 3: Shares one edge with containing triangle We can charge: The flipped edge An unshared triangle edge An unshared free edge A superfluous free edge

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 14 / 21

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SLIDE 36

Summary

Any triangulation can be made 4-connected by 3n−7

5

  • flips

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 15 / 21

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SLIDE 37

Lower Bound

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 16 / 21

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SLIDE 38

Lower Bound

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 16 / 21

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SLIDE 39

Lower Bound

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 16 / 21

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SLIDE 40

Lower Bound

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 16 / 21

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SLIDE 41

Lower Bound

(3n − 10)/5 edge-disjoint separating triangles

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 16 / 21

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SLIDE 42

Summary

Any triangulation can be made 4-connected by 3n−7

5

  • flips

There are triangulations where this requires 3n−10

5

  • flips

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 17 / 21

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SLIDE 43

Summary

n − 4 2n − 11 Total: 6n − 30 4-connected

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 18 / 21

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SLIDE 44

Summary

n − 4 2n − 11 Total: 6n − 30 3n−7

5

  • 5.2n − 24.8

4-connected

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 18 / 21

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SLIDE 45

Summary

n − 4 2n − 11 Total: 6n − 30 3n−7

5

  • 5.2n − 24.8

4-connected ≥ 3n−10

5

  • Sander Verdonschot (Carleton University)

Making Triangulations 4-connected December 19, 2011 18 / 21

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SLIDE 46

Summary

n − 4 2n − 11 Total: 6n − 30 3n−7

5

  • 5.2n − 24.8

4-connected ≥ 3n−10

5

  • ≥ 2n − 15

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 18 / 21

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SLIDE 47

Summary

n − 4 2n − 11 Total: 6n − 30 3n−7

5

  • 5.2n − 24.8

4-connected ≥ 3n−10

5

  • ≥ 2n − 15

2n − 15 5.2n − 32.8

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 18 / 21

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SLIDE 48

Summary

n − 4 2n − 11 Total: 6n − 30 3n−7

5

  • 5.2n − 24.8

4-connected ≥ 3n−10

5

  • ≥ 2n − 15

2n − 15 5.2n − 32.8 ≥ 2n − 15

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 18 / 21

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SLIDE 49

The End

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 19 / 21

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SLIDE 50

Which edges to flip?

Case 4: Shares an edge with containing triangle and one with non-containing triangle We can charge: The flipped edge An unshared triangle edge An unshared free edge A superfluous free edge

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 20 / 21

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SLIDE 51

Which edges to flip?

Case 5: Shares an edge with containing triangle and two with non-containing triangles We can charge: The flipped edge An unshared triangle edge An unshared free edge A superfluous free edge

Sander Verdonschot (Carleton University) Making Triangulations 4-connected December 19, 2011 21 / 21