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Coding, counting, and sampling triangulations and other planar - - PowerPoint PPT Presentation

Nov 12, 2023 248 likes •1.65k views

. Coding, counting, and sampling triangulations and other planar graphs Gilles S CHAEFFER CNRS, cole Polytechnique A N OVERVIEW OF THE TALK I. 3-c planar graphs II. Binary trees and a combinatorial approach III. From trees to dissections,

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SLIDE 1

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Coding, counting, and sampling triangulations and other planar graphs

Gilles SCHAEFFER CNRS, École Polytechnique

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SLIDE 2

AN OVERVIEW OF THE TALK

  • I. 3-c planar graphs
  • II. Binary trees and a combinatorial

approach

  • III. From trees to dissections,

counting and sampling.

  • IV. Minimal α-orientations, coding.
  • V. Trees and orientations everywhere.
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SLIDE 3

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Part 1. Some combinatorial structures.

CASTING for this part : 3-connected planar graphs polyhedral graphs, irreducible dissections lion, triceratops

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3-CONNECTED PLANAR GRAPHS

A planar graph is 3-connected if there is no 2-separator.

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SLIDE 5

3-CONNECTED PLANAR GRAPHS

A planar graph is 3-connected if there is no 2-separator. Whitney : 3-connected planar graphs have a unique embedding up to homeomorphisms of the non-oriented sphere, i.e. a unique planar map. 2-separators give raise to different maps for the same graph

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SLIDE 6

3-CONNECTED PLANAR GRAPHS

A planar graph is 3-connected if there is no 2-separator. Whitney : 3-connected planar graphs have a unique embedding up to homeomorphisms of the non-oriented sphere, i.e. a unique planar map. 2-separators give raise to different maps for the same graph

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SLIDE 7

3-CONNECTED PLANAR GRAPHS

A planar graph is 3-connected if there is no 2-separator. Whitney : 3-connected planar graphs have a unique embedding up to homeomorphisms of the non-oriented sphere, i.e. a unique planar map. 2-separators give raise to different maps for the same graph

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SLIDE 8

3-CONNECTED PLANAR GRAPHS

A planar graph is 3-connected if there is no 2-separator. Whitney : 3-connected planar graphs have a unique embedding up to homeomorphisms of the non-oriented sphere, i.e. a unique planar map. 2-separators give raise to different maps for the same graph

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SLIDE 9

3-CONNECTED PLANAR GRAPHS

A planar graph is 3-connected if there is no 2-separator. Whitney : 3-connected planar graphs have a unique embedding up to homeomorphisms of the non-oriented sphere, i.e. a unique planar map. 2-separators give raise to different maps for the same graph Steinitz : They are the 2-skeletons of 3d convex polyhedra.

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SLIDE 10

WHAT DO WE DO WITH 3-CONNECTED PLANAR GRAPHS ?

We want to count them : Tutte counted rooted 3-c planar maps in the 60’s, according to their number of edges, Mullin and Schellenberg according to the numbers of faces and vertices. We want to generate them uniformly at random : ⇒ random triangulations and random combinatorial planar maps in general are popular models of discrete random surfaces in physics : random sampler are used to make “experiments” about “2d quantum gravity” (Ambjorn et al. 94,...). ⇒ random graphs are sometimes used to test graph drawing algorithms. ⇒ uniform 3-connected planar graphs are needed to sample labelled planar graphs uniformly (Bodirsky–Gröpl–Kang 03) We want to encode them compactly.

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SLIDE 11

WHAT DO WE DO WITH 3-CONNECTED PLANAR GRAPHS ?

3-connected planar maps = the standard abstraction of the combinatorial part

  • f polygonal meshes with spherical topology (half-edge representations...)

⇒ a number of compression algorithms improving compression rates

Rossignac’s Edgebreaker (98), Touma-Gotsman valency coder (99)...

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SLIDE 12

PLANAR MAPS AND DISSECTIONS

Start from a planar map

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SLIDE 13

PLANAR MAPS AND DISSECTIONS

Start from a planar map Triangulate faces from new black vertices

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SLIDE 14

PLANAR MAPS AND DISSECTIONS

Start from a planar map Triangulate faces from new black vertices

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SLIDE 15

PLANAR MAPS AND DISSECTIONS

Start from a planar map Triangulate faces from new black vertices Forget former edges ⇒ quadrangles

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SLIDE 16

PLANAR MAPS AND DISSECTIONS

Start from a planar map Triangulate faces from new black vertices Forget former edges ⇒ quadrangles ⇒ a quadrangular dissection

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SLIDE 17

PLANAR MAPS AND DISSECTIONS

  • Proposition. This is one-to-one between :

3-connected planar maps with n edges, irreducible dissections with n faces. Irreducible = all 4-cycles are faces

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SLIDE 18

PLANAR MAPS AND DISSECTIONS

  • Proposition. This is one-to-one between :

3-connected planar maps with n edges, irreducible dissections with n faces. Irreducible = all 4-cycles are faces

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SLIDE 19

PLANAR MAPS AND DISSECTIONS

  • Proposition. This is one-to-one between :

3-connected planar maps with n edges, irreducible dissections with n faces. Irreducible = all 4-cycles are faces

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SLIDE 20

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Conclusion of Part 1.

Our aim : to code, count and sample 3-c planar graphs. Equivalently we can consider irreducible dissections.

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SLIDE 21

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Part 2. A combinatorial approach to counting, coding and sampling.

CASTING for this part : binary trees Catalan numbers INSPIRATION for this part : Rémi, Łukasiewicz, folklore...

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SLIDE 22

BINARY TREES...

Let Bn be the set of binary trees with n inner nodes. Well known : |Bn| = 1 n + 1 2n n

  • , the nth Catalan number.

Seek a constructive proof of this formula and use it for sampling.

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SLIDE 23

BINARY TREES...

Let Bn be the set of binary trees with n inner nodes. Well known : |Bn| = 1 n + 1 2n n

  • , the nth Catalan number.

In other terms : 2(2n − 1)|Bn−1| = (n + 1)|Bn|.

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SLIDE 24

BINARY TREES...

Let Bn be the set of binary trees with n inner nodes. A tree of Bn has n + 1 leaves. Well known : |Bn| = 1 n + 1 2n n

  • , the nth Catalan number.

In other terms : 2(2n − 1)|Bn−1| = (n + 1)|Bn|.

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SLIDE 25

BINARY TREES...

Let Bn be the set of binary trees with n inner nodes. A tree of Bn has n + 1 leaves. Well known : |Bn| = 1 n + 1 2n n

  • , the nth Catalan number.

In other terms : 2(2n − 1)|Bn−1| = |{leaves} × Bn|.

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SLIDE 26

BINARY TREES...

Let Bn be the set of binary trees with n inner nodes. A tree of Bn has n + 1 leaves. A tree of Bn−1 has 2n − 1 edges. (including the root edge) Well known : |Bn| = 1 n + 1 2n n

  • , the nth Catalan number.

In other terms : 2(2n − 1)|Bn−1| = |{leaves} × Bn|.

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SLIDE 27

BINARY TREES...

Let Bn be the set of binary trees with n inner nodes. A tree of Bn has n + 1 leaves. A tree of Bn−1 has 2n − 1 edges. (including the root edge) Well known : |Bn| = 1 n + 1 2n n

  • , the nth Catalan number.

In other terms : |{l, r} × {edges} × Bn−1| = |{leaves} × Bn|.

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SLIDE 28

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting.

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SLIDE 29

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 30

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 31

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 32

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 33

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 34

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 35

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 36

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 37

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 38

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 39

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 40

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 41

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 42

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 43

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 44

BINARY TREES... CONSTRUCTIVE COUNTING

A bijection : {l, r} × {edges} × Bn−1 ↔ {leaves} × Bn. This yields : A proof of the recurrence ⇒ constructive counting. A random sampling algorithm : Pick a side in {l, r} and an edge uniformly at random and grow ⇒ the nth tree is uniform in Bn.

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SLIDE 45

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 1

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SLIDE 46

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 11

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SLIDE 47

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 111

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SLIDE 48

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 1110

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SLIDE 49

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 11101

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SLIDE 50

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 111010

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SLIDE 51

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 1110100

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SLIDE 52

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 11101000

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SLIDE 53

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 111010001

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SLIDE 54

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 1110100011

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SLIDE 55

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 11101000110

slide-56
SLIDE 56

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 111010001100

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SLIDE 57

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 1110100011001

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SLIDE 58

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 11101000110011

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SLIDE 59

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 111010001100110

slide-60
SLIDE 60

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 1110100011001100

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SLIDE 61

BINARY TREES... CONSTRUCTIVE COUNTING

Another observation : |Bn| = 1 n + 1 2n n

  • ∼ 22n · cn−5/2.

⇒ It should be possible to encode trees of |Bn| by words of {0, 1}2n. This can be done by prefix encoding. – Write 1 for left edges, 0 for right ones along a prefix traversal. 1110100011001100 This code has length 2n and is optimal in the sense that a code must use at least 2n + o(n) bits in the worst case.

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SLIDE 62

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Conclusion of Part 2.

Bijections can help for counting, coding and sampling. Binary trees are well known...

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SLIDE 63

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Part 3. The closure of a binary tree into a dissection

CASTING for this part : binary trees (again) irreducible dissections (stunt men return) 3-connected planar graphs (hors champs)

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SLIDE 64

TUTTE’S RESULTS ABOUT 3-CONNECTED PLANAR GRAPHS

The number 3-connected planar graphs ? Tutte (62) : a complicated formula for rooted 3-c planar maps. However these numbers are “Catalan related” (their generating function lies in the same algebraic extention). ⇒ explain this combinatorially... We would like to find a simple one-to-one correspondence between 3c planar graphs and binary trees.

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SLIDE 65

THE CLOSURE OF A BINARY TREE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n + 1 edges. Compare to : 3-c planar graphs : n edges, i vertices, j faces, with i + j = n + 2 (Euler).

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SLIDE 66

THE CLOSURE OF A BINARY TREE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n + 1 edges. Compare to : Dissections : n faces, 2n edges and n + 2 vertices (by Euler).

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SLIDE 67

THE CLOSURE OF A BINARY TREE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n + 1 edges. Compare to : Dissections : n faces, 2n edges and n + 2 vertices (by Euler). Create faces of degree four, keeping the number of vertices and edges...

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SLIDE 68

THE CLOSURE OF A BINARY TREE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n + 1 edges. Compare to : Dissections : n faces, 2n edges and n + 2 vertices (by Euler). Create faces of degree four, keeping the number of vertices and edges... Local closure : close an leaf followed in ccw order by 3 sides of inner edges.

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SLIDE 69

THE CLOSURE OF A BINARY TREE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n + 1 edges. Compare to : Dissections : n faces, 2n edges and n + 2 vertices (by Euler). Create faces of degree four, keeping the number of vertices and edges... Local closure : close an leaf followed in ccw order by 3 sides of inner edges.

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SLIDE 70

THE CLOSURE OF A BINARY TREE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n + 1 edges. Compare to : Dissections : n faces, 2n edges and n + 2 vertices (by Euler). Create faces of degree four, keeping the number of vertices and edges... Local closure : close an leaf followed in ccw order by 3 sides of inner edges.

slide-71
SLIDE 71

THE CLOSURE OF A BINARY TREE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n + 1 edges. Compare to : Dissections : n faces, 2n edges and n + 2 vertices (by Euler). Create faces of degree four, keeping the number of vertices and edges... Local closure : close an leaf followed in ccw order by 3 sides of inner edges.

slide-72
SLIDE 72

THE CLOSURE OF A BINARY TREE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n + 1 edges. Compare to : Dissections : n faces, 2n edges and n + 2 vertices (by Euler). Create faces of degree four, keeping the number of vertices and edges... Local closure : close an leaf followed in ccw order by 3 sides of inner edges. Remark : local closures commute, the resulting partial closure is well defined.

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SLIDE 73

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6.

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SLIDE 74

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6. Complete closure : The remaining leaves can be attached to the vertices of hexagon so as to form faces of degree 4.

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SLIDE 75

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6. Complete closure : The remaining leaves can be attached to the vertices of hexagon so as to form faces of degree 4.

slide-76
SLIDE 76

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6. Complete closure : The remaining leaves can be attached to the vertices of hexagon so as to form faces of degree 4.

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SLIDE 77

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6. Complete closure : The remaining leaves can be attached to the vertices of hexagon so as to form faces of degree 4.

slide-78
SLIDE 78

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6. Complete closure : The remaining leaves can be attached to the vertices of hexagon so as to form faces of degree 4.

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SLIDE 79

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6. Complete closure : The remaining leaves can be attached to the vertices of hexagon so as to form faces of degree 4.

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SLIDE 80

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6. Complete closure : The remaining leaves can be attached to the vertices of hexagon so as to form faces of degree 4.

slide-81
SLIDE 81

BINARY TREES AND THE CLOSURE

Consider a tree of |Bn| : — n inner vertices, — n − 1 inner edges, — n + 2 leaves (root included), — and 2n edges. Partial closure : when all local closures are done the numbers k of remaining leaves and ℓ of sides of inner edges in the infinite face satisfy 2k − ℓ = 6. Complete closure : The remaining leaves can be attached to the vertices of hexagon so as to form faces of degree 4. Up to rotation of the hexagon there is a unique way to do this.

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SLIDE 82

BINARY TREES AND DISSECTION OF A HEXAGON

Theorem (Fusy–Poulalhon–Schaeffer 04). Closure is a bijection between unrooted binary trees with n inner nodes irreducible dissection of an hexagon with n internal vertices.

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SLIDE 83

BINARY TREES AND DISSECTION OF A HEXAGON

Theorem (Fusy–Poulalhon–Schaeffer 04). Closure is a bijection between unrooted binary trees with n inner nodes irreducible dissection of an hexagon with n internal vertices. Impossibility of 4-cycle is checked by a counting argument.

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SLIDE 84

BINARY TREES AND DISSECTION OF A HEXAGON

Theorem (Fusy–Poulalhon–Schaeffer 04). Closure is a bijection between unrooted binary trees with n inner nodes irreducible dissection of an hexagon with n internal vertices. Impossibility of 4-cycle is checked by a counting argument. Orient half-edges of the tree away from a vertex v inside.

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SLIDE 85

BINARY TREES AND DISSECTION OF A HEXAGON

Theorem (Fusy–Poulalhon–Schaeffer 04). Closure is a bijection between unrooted binary trees with n inner nodes irreducible dissection of an hexagon with n internal vertices. Impossibility of 4-cycle is checked by a counting argument. Orient half-edges of the tree away from a vertex v inside. Count outgoing half-edges inside the 4-cycle : 3 + 2k = 2(k + 1)

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SLIDE 86

BINARY TREES AND DISSECTION OF A HEXAGON

Theorem (Fusy–Poulalhon–Schaeffer 04). Closure is a bijection between unrooted binary trees with n inner nodes irreducible dissection of an hexagon with n internal vertices.

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SLIDE 87

BINARY TREES AND DISSECTION OF A HEXAGON

Theorem (Fusy–Poulalhon–Schaeffer 04). Closure is a bijection between unrooted binary trees with n inner nodes irreducible dissection of an hexagon with n internal vertices. Corollaries : The number of rooted dissections of a hexagon with n inner vertices is 6 n + 2 1 n + 1 2n n

  • .
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SLIDE 88

BINARY TREES AND DISSECTION OF A HEXAGON

Theorem (Fusy–Poulalhon–Schaeffer 04). Closure is a bijection between unrooted binary trees with n inner nodes irreducible dissection of an hexagon with n internal vertices. Corollaries : The number of rooted dissections of a hexagon with n inner vertices is 6 n + 2 1 n + 1 2n n

  • .

Encode a dissection of the hexagon by the 2n bits coding the tree.

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SLIDE 89

BINARY TREES AND DISSECTION OF A HEXAGON

Theorem (Fusy–Poulalhon–Schaeffer 04). Closure is a bijection between unrooted binary trees with n inner nodes irreducible dissection of an hexagon with n internal vertices. Corollaries : The number of rooted dissections of a hexagon with n inner vertices is 6 n + 2 1 n + 1 2n n

  • .

Sample uniform random rooted dissections in linear time.

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SLIDE 90

DISSECTION OF A HEXAGON OR OF A SQUARE

Consider an irreducible dissection associated with a 3-c planar graph. Removing one edge yields an irreducible dissection of a hexagon. ⇒ our approach thus immediately yields a code for 3-c planar graphs. this code is “optimal” again..

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SLIDE 91

DISSECTION OF A HEXAGON OR OF A SQUARE

Consider an irreducible dissection associated with a 3-c planar graph. Removing one edge yields an irreducible dissection of a hexagon. ⇒ our approach thus immediately yields a code for 3-c planar graphs. this code is “optimal” again..

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SLIDE 92

DISSECTION OF A HEXAGON OR OF A SQUARE

Consider an irreducible dissection associated with a 3-c planar graph. Removing one edge yields an irreducible dissection of a hexagon. ⇒ our approach thus immediately yields a code for 3-c planar graphs. this code is “optimal” again..

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SLIDE 93

DISSECTION OF A HEXAGON OR OF A SQUARE

Consider an irreducible dissection associated with a 3-c planar graph. Removing one edge yields an irreducible dissection of a hexagon. ⇒ our approach thus immediately yields a code for 3-c planar graphs. this code is “optimal” again..

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SLIDE 94

DISSECTION OF A HEXAGON OR OF A SQUARE

Consider an irreducible dissection associated with a 3-c planar graph. Removing one edge yields an irreducible dissection of a hexagon. ⇒ our approach thus immediately yields a code for 3-c planar graphs. this code is “optimal” again.. Conversely : an irreducible dissection of a hexagon ⇒ an irreducible dissection of a square iff there was not a diagonal of length 3.

slide-95
SLIDE 95

DISSECTION OF A HEXAGON OR OF A SQUARE

Consider an irreducible dissection associated with a 3-c planar graph. Removing one edge yields an irreducible dissection of a hexagon. ⇒ our approach thus immediately yields a code for 3-c planar graphs. this code is “optimal” again.. Conversely : an irreducible dissection of a hexagon ⇒ an irreducible dissection of a square iff there was not a diagonal of length 3.

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SLIDE 96

DISSECTION OF A HEXAGON OR OF A SQUARE

Consider an irreducible dissection associated with a 3-c planar graph. Removing one edge yields an irreducible dissection of a hexagon. ⇒ our approach thus immediately yields a code for 3-c planar graphs. this code is “optimal” again.. Conversely : an irreducible dissection of a hexagon ⇒ an irreducible dissection of a square iff there was not a diagonal of length 3. ⇒ sampling by rejection : try to add an edge, restart from scratch if not ok.

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SLIDE 97

.

Conclusion of Part 3.

{ 3-connected planar graphs } ≈ { Irreducible dissections of a hexagon } ≡ { binary trees }. Corrollaries : — a formula for rooted irreducible dissections, — a linear time random sampler for 3-c planar graphs, ⇒ improvment for the generator for planar graphs of Bodirsky et al. — and a compact code for polyhedral meshes with spherical topology.

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SLIDE 98

.

Conclusion of Part 3.

{ 3-connected planar graphs } ≈ { Irreducible dissections of a hexagon } ≡ { binary trees }. Corrollaries : — a formula for rooted irreducible dissections, — a linear time random sampler for 3-c planar graphs, ⇒ improvment for the generator for planar graphs of Bodirsky et al. — and a compact code for polyhedral meshes with spherical topology. but until now I did not show how to compute the code.

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SLIDE 99

.

Part 4. Minimal α-orientations and coding. (a glimpse of the machinery behind)

CASTING for this part :

  • rientations

derived map INSPIRATION for this part de Frayssex, Ossona de Mendez, Brehm, Felsner...

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SLIDE 100

RETURN TO THE MAIN THEOREM...

Closure is a bijection between unrooted binary trees with n inner nodes and irreducible dissection of an hexagon with n internal vertices. Orient all half-edges of the binary tree ⇒ an “orientation” of the dissection.

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SLIDE 101

RETURN TO THE MAIN THEOREM...

Closure is a bijection between unrooted binary trees with n inner nodes and irreducible dissection of an hexagon with n internal vertices. Orient all half-edges of the binary tree ⇒ an “orientation” of the dissection. All internal vertices have out-degree 3.

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SLIDE 102

RETURN TO THE MAIN THEOREM...

Closure is a bijection between unrooted binary trees with n inner nodes and irreducible dissection of an hexagon with n internal vertices. Orient all half-edges of the binary tree ⇒ an “orientation” of the dissection. All internal vertices have out-degree 3.

  • Proposition. By construction, there are no cw circuits.
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SLIDE 103

RETURN TO THE MAIN THEOREM...

Closure is a bijection between unrooted binary trees with n inner nodes and irreducible dissection of an hexagon with n internal vertices. Orient all half-edges of the binary tree ⇒ an “orientation” of the dissection. All internal vertices have out-degree 3.

  • Proposition. By construction, there are no cw circuits.

Conversely, in a “3-orientation” without cw circuit, edges →← form a tree.

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SLIDE 104

RETURN TO THE MAIN THEOREM...

Refined Theorem : Closure is a bijection between unrooted binary trees and irreducible dissections of a hexagon without cw circuits. Orient all half-edges of the binary tree ⇒ an “orientation” of the dissection. All internal vertices have out-degree 3.

  • Proposition. By construction, there are no cw circuits.

Conversely, in a “3-orientation” without cw circuit, edges →← form a tree.

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SLIDE 105

α-ORIENTATIONS

Let α be an out-degree prescription for the vertices of a planar graph. α-orientation = orientation of edges respecting α. Theorem (Felsner 03, Ossona de Mendez 94) If there exists an α-orientation, then the transformation return a cw circuit defines a distributive lattice on the set of α-orientation. In particular : the minimal α-orientation is the only α-orientation without cw circuits.

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SLIDE 106

α-ORIENTATIONS AND DISSECTIONS

The theory does not directly apply to us : we have doubly

  • riented edges.
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SLIDE 107

α-ORIENTATIONS AND DISSECTIONS

The theory does not directly apply to us : we have doubly

  • riented edges.

“3-oriented” dissection ⇔ α-oriented derived map : α(◦) = 3, α(×) = 1.

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SLIDE 108

α-ORIENTATIONS AND DISSECTIONS

The theory does not directly apply to us : we have doubly

  • riented edges.

“3-oriented” dissection ⇔ α-oriented derived map : α(◦) = 3, α(×) = 1. prove : without cw circuits ⇔ without cw circuits apply Felsner’s theorem to the derived map. ⇒ this proves that the closure send bijectively trees on dissections.

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SLIDE 109

CONSTRUCTING THE MINIMAL α-ORIENTATIONS

For coding, we still need to show that one can construct the minimal

  • rientation in linear time.

The construction is akin to the construction for mini- mal 3-orientations of trian- gulations (Kant, Brehm). The base line a2a3 is fixed. The rightmost nonsepara- ting active vertex on the frontier is removed and in- cident edges are oriented.

a3 a2 a1

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SLIDE 110

CONSTRUCTING THE MINIMAL α-ORIENTATIONS

For coding, we still need to show that one can construct the minimal

  • rientation in linear time.

The construction is akin to the construction for mini- mal 3-orientations of trian- gulations (Kant, Brehm). The base line a2a3 is fixed. The rightmost nonsepara- ting active vertex on the frontier is removed and in- cident edges are oriented.

a1 a3 a2

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SLIDE 111

CONSTRUCTING THE MINIMAL α-ORIENTATIONS

For coding, we still need to show that one can construct the minimal

  • rientation in linear time.

The construction is akin to the construction for mini- mal 3-orientations of trian- gulations (Kant, Brehm). The base line a2a3 is fixed. The rightmost nonsepara- ting active vertex on the frontier is removed and in- cident edges are oriented.

a1 a3 a2

slide-112
SLIDE 112

CONSTRUCTING THE MINIMAL α-ORIENTATIONS

For coding, we still need to show that one can construct the minimal

  • rientation in linear time.

The construction is akin to the construction for mini- mal 3-orientations of trian- gulations (Kant, Brehm). The base line a2a3 is fixed. The rightmost nonsepara- ting active vertex on the frontier is removed and in- cident edges are oriented.

a1 a3 a2

slide-113
SLIDE 113

CONSTRUCTING THE MINIMAL α-ORIENTATIONS

For coding, we still need to show that one can construct the minimal

  • rientation in linear time.

The construction is akin to the construction for mini- mal 3-orientations of trian- gulations (Kant, Brehm). The base line a2a3 is fixed. The rightmost nonsepara- ting active vertex on the frontier is removed and in- cident edges are oriented.

a1 a3 a2

slide-114
SLIDE 114

CONSTRUCTING THE MINIMAL α-ORIENTATIONS

For coding, we still need to show that one can construct the minimal

  • rientation in linear time.

The construction is akin to the construction for mini- mal 3-orientations of trian- gulations (Kant, Brehm). The base line a2a3 is fixed. The rightmost nonsepara- ting active vertex on the frontier is removed and in- cident edges are oriented. Theorem (FPS04). This process constructs the minimal α-orientation.

a1 a3 a2

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SLIDE 115

THE COMPLETE ENCODING PROCEDURE

Complete the 3-c graph to make it canonical.

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SLIDE 116

THE COMPLETE ENCODING PROCEDURE

Complete the 3-c graph to make it canonical.

a3 a2 a1

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SLIDE 117

THE COMPLETE ENCODING PROCEDURE

Complete the 3-c graph to make it canonical. Superimpose the dual.

a3 a2 a1

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SLIDE 118

THE COMPLETE ENCODING PROCEDURE

Complete the 3-c graph to make it canonical. Superimpose the dual. Orient the derived map.

a1 a3 a2

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SLIDE 119

THE COMPLETE ENCODING PROCEDURE

Complete the 3-c graph to make it canonical. Superimpose the dual. Orient the derived map. Transport orientation to the dissection.

a1 a3 a2

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SLIDE 120

THE COMPLETE ENCODING PROCEDURE

Complete the 3-c graph to make it canonical. Superimpose the dual. Orient the derived map. Transport orientation to the dissection. Detach simply oriented edges.

slide-121
SLIDE 121

THE COMPLETE ENCODING PROCEDURE

Complete the 3-c graph to make it canonical. Superimpose the dual. Orient the derived map. Transport orientation to the dissection. Detach simply oriented edges.

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SLIDE 122

.

Conclusion of Part 4.

α-orientations play a key role in proofs. “optimal” encoding can be performed in linear time.

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SLIDE 123

.

Part 5. Other instances.

CASTING for this part : Triangulations and Schnyder trees [Poulalhon-Schaeffer] Eulerian maps and their balanced orientations [Fusy] Simple quadrangular dissections and 1-2-orientations [Fusy-Poulalhon]

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SLIDE 124

TRIANGULATIONS

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SLIDE 125

TRIANGULATIONS

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SLIDE 126

TRIANGULATIONS

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SLIDE 127

TRIANGULATIONS

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SLIDE 128

TRIANGULATIONS

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SLIDE 129

TRIANGULATIONS

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SLIDE 130

TRIANGULATIONS

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SLIDE 131

TRIANGULATIONS

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SLIDE 132

TRIANGULATIONS

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SLIDE 133

TRIANGULATIONS

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SLIDE 134

TRIANGULATIONS

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SLIDE 135

TRIANGULATIONS

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SLIDE 136

TRIANGULATIONS

− → Theorem (Poulalhon-Schaeffer 03). This is a bijection and its inverse is based on the minimal 3-orientations of a triangulation.

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SLIDE 137

.

Conclusion of Part 5.

Minimal α-orientations hide trees... It remains to give a common explanation to these various results : a theory of trees and minimal α-orientations.

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SLIDE 138

.

A conclusion to bring home.

Nice counting formulas must have simple interpretations Looking for these reveals hidden combinatorial structure