Encoding Toroidal Triangulations - Vincent DESPRÉ

Nov 5, 2015 - + Fix a face distributive lattice. Related to older results of Propp (1993), Ossona de. Mendez (1994), Felsner (2004). 11 / 16. Page 79. Encoding.
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Encoding Toroidal Triangulations Vincent DESPRE Planar Case 3-Orientations Algorithm

Encoding Toroidal Triangulations

Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Vincent Despré, Daniel Goncalves and Benjamin Lévêque

Torus Case Algorithm Structure Encoding

gipsa-lab, G-SCOP, Grenoble [email protected]

Future Sight

5 November 2015

1 / 16

Encoding Toroidal Triangulations

Planar triangulations

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

2 / 16

Encoding Toroidal Triangulations

Planar triangulations

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Euler : v − e + f = 2 Triangulation : 3f = 2e

=⇒

e (e − 3) eint

= = =

3v − 6 3(v − 3) 3vint

2 / 16

Encoding Toroidal Triangulations

Planar triangulations

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Euler : v − e + f = 2 Triangulation : 3f = 2e

=⇒

e (e − 3) eint

= = =

3v − 6 3(v − 3) 3vint

Associate to each internal vertex three incident edges 2 / 16

Encoding Toroidal Triangulations

Planar triangulations

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Euler : v − e + f = 2 Triangulation : 3f = 2e

=⇒

e (e − 3) eint

= = =

3v − 6 3(v − 3) 3vint

Associate to each internal vertex three incident edges 2 / 16

Encoding Toroidal Triangulations

Planar triangulations

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Euler : v − e + f = 2 Triangulation : 3f = 2e

=⇒

e (e − 3) eint

= = =

3v − 6 3(v − 3) 3vint

Associate to each internal vertex three incident edges 2 / 16

Encoding Toroidal Triangulations

Planar triangulations

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Euler : v − e + f = 2 Triangulation : 3f = 2e

=⇒

e (e − 3) eint

= = =

3v − 6 3(v − 3) 3vint

Associate to each internal vertex three incident edges 2 / 16

Encoding Toroidal Triangulations

Planar triangulations

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Euler : v − e + f = 2 Triangulation : 3f = 2e

=⇒

e (e − 3) eint

= = =

3v − 6 3(v − 3) 3vint

Associate to each internal vertex three incident edges and deduce a 3-orientation 2 / 16

Encoding Toroidal Triangulations

Schnyder Woods

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Schnyder wood

3 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Encoding - Poulalhon, Schaeffer (2003)

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

4 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Bijection

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

5 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Distributive lattice

Future Sight

6 / 16

Encoding Toroidal Triangulations

Conditions

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Distributive lattice

Future Sight

6 / 16

Encoding Toroidal Triangulations

Encoding the Tree

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

0 1

Encoding Summary

1 0

1

Higher Genus

0 0

1

0

Orientations Structure

1

Summary

1 0 Torus Case Algorithm

0

1

1

Structure Encoding

Future Sight

0 0 1

1

1 0

0

0

1

0

0 1

1

1

0

101101011101011010001010001100

7 / 16

Encoding Toroidal Triangulations

Encoding the Tree

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

0 1

Encoding Summary

1 0

1

Higher Genus

0 0

1

0

Orientations Structure

1

Summary

1 0 Torus Case Algorithm

0

1

1

Structure Encoding

Future Sight

0 0 1

1

1 0

0

0

1

0

0 1

1

1

0

101101011101011010001010001100

6n bits

7 / 16

Encoding Toroidal Triangulations

Encoding the Tree

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

0 1

Encoding Summary

1 0

1

Higher Genus

0 0

1

0

Orientations Structure

1

Summary

1 0 Torus Case Algorithm

0

1

1

Structure Encoding

Future Sight

0 0 1

1

1 0

0

0

1

0

0 1

1

1

0

101101011101011010001010001100

6n bits

7 / 16

Encoding Toroidal Triangulations

Encoding the Tree

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

0 1

Encoding Summary

1 0

1

Higher Genus

0 0

1

0

Orientations Structure

1

Summary

1 0 Torus Case Algorithm

0

1

1

Structure Encoding

Future Sight

0 0 1

1

1 0

0

0

1

0

0 1

1

1

0

101101011101011010001010001100

7 / 16

Encoding Toroidal Triangulations

Encoding the Tree

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

0 1

Encoding Summary

1 0

1

Higher Genus

0 0

1

0

Orientations Structure

1

Summary

1 0 Torus Case Algorithm

0

1

1

Structure Encoding

Future Sight

0 0 1

1

1 0

0

0

1

0

0 1

1

1

0

1010011001000000001100

7 / 16

Encoding Toroidal Triangulations

Encoding the Tree

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

0 1

Encoding Summary

1 0

1

Higher Genus

0 0

1

0

Orientations Structure

1

Summary

1 0 Torus Case Algorithm

0

1

1

Structure Encoding

Future Sight

0 0 1

1

1 0

0

0

1

0

0 1

1

1

0

1010011001000000001100

4n bits

7 / 16

Encoding Toroidal Triangulations

Encoding the Tree

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

0 1

Encoding Summary

1 0

1

Higher Genus

0 0

1

0

Orientations Structure

1

Summary

1 0 Torus Case Algorithm

0

1

1

Structure Encoding

Future Sight

0 0 1

1

1 0

0

0

1

0

0 1

1

1

0

1010011001000000001100

4n bits (n×1)

7 / 16

Encoding Toroidal Triangulations

Encoding the Tree

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

0 1

Encoding Summary

1 0

1

Higher Genus

0 0

1

0

Orientations Structure

1

Summary

1 0 Torus Case Algorithm

0

1

1

Structure Encoding

Future Sight

0 0 1

1

1 0

0

0

1

0

0 1

1

1

0

1010011001000000001100 4n bits (n×1) OPTIMAL !

3,25n bits

7 / 16

Encoding Toroidal Triangulations Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

Steps of the Algorithm In addition it gives a new proof of Tutte’s formula: Tn+2 =

2(4n−3)! n!(3n−1)!

log(Tn+2 ) ∼ log( 256 27 )n ' 3.25n

Encoding Summary

Higher Genus Orientations Structure Summary

INPUT: A triangulation of the plane. OUTPUT: The corresponding encoding.

Torus Case Algorithm Structure Encoding

Future Sight

i/ Choose an orientation. ii/ Go down in the lattice to find the minimum element. iii/ Run Poulalhon and Schaeffer’s algorithm. iv/ Encode.

8 / 16

Encoding Toroidal Triangulations Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

Triangulations on Oriented Surfaces Euler’s formula in genus g : v − e + f = 2 − 2g Double Torus Plane Torus

Genus 0 1

Triangulation e = 3v − 6 e = 3v

Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

9 / 16

Encoding Toroidal Triangulations Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

Triangulations on Oriented Surfaces Euler’s formula in genus g : v − e + f = 2 − 2g Double Torus Plane Torus

Genus 0 1

Triangulation e = 3v − 6 e = 3v

Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

9 / 16

Encoding Toroidal Triangulations Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

Triangulations on Oriented Surfaces Euler’s formula in genus g : v − e + f = 2 − 2g Double Torus Plane Torus

Genus 0 1

Triangulation e = 3v − 6 e = 3v

Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

9 / 16

Encoding Toroidal Triangulations Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

Triangulations on Oriented Surfaces Euler’s formula in genus g : v − e + f = 2 − 2g Double Torus Plane Torus

Genus 0 1

Triangulation e = 3v − 6 e = 3v

Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

9 / 16

Encoding Toroidal Triangulations Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection

Triangulations on Oriented Surfaces Euler’s formula in genus g : v − e + f = 2 − 2g Double Torus Plane Torus

Genus 0 1

Triangulation e = 3v − 6 e = 3v

Encoding Summary

Higher Genus Orientations Structure

Theorem (Barát and Thomassen 2006)

Summary

Torus Case Algorithm Structure

Triangulation on a surface =⇒ orientation of the edges such that d+ (v) = 0 mod 3.

Encoding

Future Sight

Theorem (Albar, Gonçalves and Knauer 2014) Triangulation on a surface g ≥ 1 =⇒ orientation of the edges such that d+ (v) = 0 mod 3, d+ (v) > 0

9 / 16

Encoding Toroidal Triangulations Vincent DESPRE Planar Case

Characterization g = 0 : Schnyder wood ⇐⇒ 3-orientation g > 0 : Schnyder wood ⇐⇒ (0 mod 3)-orientation ? C

3-Orientations Algorithm Bijection

False !

Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

γ(C) = #→ − #←

Theorem (Gonçalves, Knauer and Lévêque 2014) Schnyder wood ⇐⇒ (0 mod 3)-orientation and for every cycle C, γ(C) = 0 mod 3 10 / 16

Encoding Toroidal Triangulations Vincent DESPRE Planar Case 3-Orientations Algorithm

Lattice structure Theorem (Gonçalves, Knauer and Lévêque 2015) Homologous orientations of a map on an orientable surface + Fix a face distributive lattice

Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Related to older results of Propp (1993), Ossona de Mendez (1994), Felsner (2004) 11 / 16

Encoding Toroidal Triangulations

Steps of the Algorithm

Vincent DESPRE Planar Case 3-Orientations Algorithm

INPUT: A triangulation of the torus. OUTPUT: The corresponding encoding.

Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

i/ Choose an orientation. ii/ Go in the good lattice. iii/ Go down in the lattice to find the minimum element. iv/ Run Poulalhon and Schaeffer’s algorithm. v/ Encode.

12 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 5

Vincent DESPRE

3

Planar Case 3-Orientations

7

Algorithm Bijection

1

Encoding Summary

2 Higher Genus Orientations Structure

4

Summary

Torus Case Algorithm

6

Structure Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 5

Vincent DESPRE

3

Planar Case 3-Orientations

7

Algorithm Bijection

1

Encoding Summary

2 Higher Genus Orientations Structure

4

Summary

Torus Case Algorithm

6

Structure Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 5

Vincent DESPRE

3

Planar Case 3-Orientations

7

Algorithm Bijection

1

Encoding Summary

2 Higher Genus Orientations Structure

4

Summary

Torus Case Algorithm

6

Structure Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 10

Vincent DESPRE

5 3

6

3

Planar Case 3-Orientations

7 7

Algorithm

8

Bijection

1

Encoding

1

Summary

2 Higher Genus Orientations Structure Summary

4 5

9

Torus Case Algorithm

6

2

4

Structure Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 10

Vincent DESPRE

5 3

6

3

Planar Case 3-Orientations

7 7

Algorithm

8

Bijection

1

Encoding

1

Summary

2 Higher Genus Orientations Structure Summary

4 5

9

Torus Case Algorithm

6

2

4

Structure Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 5

Vincent DESPRE

3

Planar Case 3-Orientations

7

Algorithm Bijection

1

Encoding Summary

2 Higher Genus Orientations Structure

4

Summary

Torus Case Algorithm

6

Structure Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 6

2

5

Vincent DESPRE

3

3

5 7

Planar Case 3-Orientations

4

7

Algorithm Bijection

1

2

3

Encoding Summary

2 Higher Genus

5

Orientations Structure

4

Summary

1

4

Torus Case Algorithm Structure

6

2

6

3

Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 6

2

5

Vincent DESPRE

3

3

5 7

Planar Case 3-Orientations

4

7

Algorithm Bijection

1

2

3

Encoding Summary

2 Higher Genus

5

Orientations Structure

4

Summary

1

4

Torus Case Algorithm Structure

6

2

6

3

Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 6

2

5

Vincent DESPRE

3

3

5 7

Planar Case 3-Orientations

4

7

Algorithm Bijection

1

2

3

Encoding Summary

2 Higher Genus

5

Orientations Structure

4

Summary

1

4

Torus Case Algorithm Structure

6

2

6

3

Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 6

2

5

Vincent DESPRE

3

3

5 7

Planar Case 3-Orientations

4

7

Algorithm Bijection

1

2

3

Encoding Summary

2 Higher Genus

5

Orientations Structure

4

Summary

1

4

Torus Case Algorithm Structure

6

2

6

3

Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 6

2

5

Vincent DESPRE

3

3

5 7

Planar Case 3-Orientations

4

7

Algorithm Bijection

1

2

3

Encoding Summary

2 Higher Genus

5

Orientations Structure

4

Summary

1

4

Torus Case Algorithm Structure

6

2

6

3

Encoding

Future Sight

13 / 16

Encoding Toroidal Triangulations

Poulalhon-Schaeffer’s Algorithm on the Torus 6

2

5

Vincent DESPRE

3

3

5 7

Planar Case 3-Orientations

4

7

Algorithm Bijection

1

2

3

Encoding Summary

2 Higher Genus

5

Orientations Structure

4

Summary

1

4

Torus Case Algorithm Structure

6

2

6

3

Encoding

Future Sight

Theorem (D., Gonçalves and Lévêque 2015) Applied on the minimal γ0 -Schnyder wood, Poulalhon and Schaeffer’s algorithm outputs a spanning unicellular map. 13 / 16

Encoding Toroidal Triangulations

Counter-examples

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

14 / 16

Encoding Toroidal Triangulations

Encoding 6

2 Vincent DESPRE

3

5 7

Planar Case

4

3-Orientations Algorithm Bijection Encoding

2

3

Summary

Higher Genus

5

Orientations Structure

1

Summary

4

Torus Case Algorithm

2

6

3

Structure Encoding

Future Sight

15 / 16

Encoding Toroidal Triangulations

Encoding 6

2

Vincent DESPRE

3

5 7

Planar Case

4

3-Orientations Algorithm Bijection Encoding

2

3

Summary

Higher Genus

5

Orientations Structure

1

Summary

4

Torus Case Algorithm Structure

2

6

3

Encoding

Future Sight

15 / 16

Encoding Toroidal Triangulations

Encoding 6

2

Vincent DESPRE

3 5

5

4

7

Planar Case

4

3-Orientations

3

Algorithm Bijection Encoding

2

3

7

Summary

Higher Genus

5

Orientations Structure

1

Summary

2

Torus Case Algorithm Structure

1

4

2

6

6

3

Encoding

Future Sight

15 / 16

Encoding Toroidal Triangulations

Encoding 6

2

Vincent DESPRE

3 5

5

4

7

Planar Case

4

3-Orientations

3

Algorithm Bijection Encoding

2

3

7

Summary

Higher Genus

5

Orientations Structure

1

Summary

2

Torus Case Algorithm Structure

1

4

2

6

6

3

Encoding

Future Sight

15 / 16

Encoding Toroidal Triangulations

Encoding 6

2

Vincent DESPRE

3 5

5

4

7

Planar Case

4

3-Orientations

3

Algorithm Bijection Encoding

2

3

7

Summary

Higher Genus

5

Orientations Structure

1

Summary

2

Torus Case Algorithm Structure

1

4

2

6

6

3

Encoding

Future Sight

15 / 16

Encoding Toroidal Triangulations

Encoding 6

2

Vincent DESPRE

3

5 7

Planar Case

4

3-Orientations Algorithm Bijection Encoding

3

Summary

Higher Genus Orientations Structure

1

Summary

Torus Case Algorithm Structure

2

6

3

0 1 1 0 0 1 10 1 00 1 0 1 1 0 1 2 0 10 1 0 1 0 5 1 01 0 01 1 00 4 1 0 1 10 1 0

Encoding

Future Sight

1010111011101001010010110100010101101000 6n bits

15 / 16

Encoding Toroidal Triangulations

Encoding

Vincent DESPRE

6

2

Planar Case

0

3

0

5

3-Orientations

00

7

Algorithm

4

Bijection

0 1

Encoding Summary

Higher Genus

2

3

Orientations Structure

5

Summary

1

Torus Case

0

0

1

1 0

0

0

0 0 0

1 1

4

0

Algorithm Structure Encoding

0

2

6

3

0

0 00 1 0

Future Sight

00110110000000100000010000 4n bits (n bits 1) 3,25n bits

15 / 16

Encoding Toroidal Triangulations

Encoding

Vincent DESPRE

6

2

Planar Case

0

3

0

5

3-Orientations

00

7

Algorithm

4

Bijection

0 1

Encoding Summary

Higher Genus

2

3

Orientations Structure

5

Summary

1

Torus Case

0

0

1

1 0

0

0

0 0 0

1 1

4

0

Algorithm Structure Encoding

0

2

6

3

0

0 00 1 0

Future Sight

00110110000000100000010000 4n bits (n bits 1) 3,25n bits Optimal, linear and bijective ! 15 / 16

Encoding Toroidal Triangulations

Future Sight

Vincent DESPRE Planar Case 3-Orientations Algorithm Bijection Encoding Summary

Higher Genus Orientations

õ Counting and sampling. õ Bijections for other toroidal maps : d-angulations, 3-connected maps, 4-connected triangulations, etc. õ Higher genus : Is the generalization of the γ0 property possible?

Structure Summary

Torus Case Algorithm Structure Encoding

Future Sight

Conjecture Triangulation on a surface g ≥ 1 =⇒ orientation of the edges such that d+ (v) = 0 mod 3, d+ (v) > 0 and no oriented non-contractible cycle in the dual.

16 / 16