Graph encoding: an overview

Luca Castelli Aleardi 21 april 2009, Dagstuhl

Graph encoding: an overview

algorithmics and combinatorics of (planar) graphs

LIX ´Ecole Polytechnique

Domain and problems

an overview

Triangulations and graphs

Geometric data

GIS Technology

Applications

Surface recontruction

Very large geometric data

St. Matthew (Stanford’s Digital Michelan- gelo Project, 2000)

186 millions vertices

6 Giga bytes (for storing on disk)

tens of minutes (for loading the model from disk)

David statue (Stanford’s Digital Michelan- gelo Project, 2000)

2 billions polygons

32 Giga bytes (without compression)

No existing algorithm nor data

structure for dealing with the entire

model

Research topics

Mesh compression

Compact representations of geometric data structures Geometric data structures

i ...

disk storage

Transmission

1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0

⇒ 2n bits for encoding a tree with n edges

ordered tree with n edges

balanced parenthesis word

kB

k =

≈ 2

n

Enumeration of plane trees with n edges

An example: plane trees

Enumeration and entropy of plane trees

Optimal encoding matching asymptotically the information-theory lower bound

log

kB

k = 2n + O(lg n)

1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0

⇒ 2n bits for encoding a tree with n edges

ordered tree with n edges

balanced parenthesis word

An example: plane trees

Enumeration and entropy of plane trees

with the guarantee the the encoding is still asymptotically optimal it is possible to test adjacency between vertices in O(1) time

( ( ( ( ) ) )

11 0 00 00 0 0 1 00 0 0 1 00 0 0 1 00 b1 b₂ b3 b4 b₅

5 2 4 4 5

0 3 2 3 2

D

B T 3

2

1 5

10

6 8

9

7 4

2n + o(n) bits are sufficient

) ( ) ) (( ( ) ) () ( ) ) (

(Jacobson, Focs89, Munro et Raman Focs97)

An example: succinct trees

the encoding is still asymptotically optimal in the worst case

it is possible to match the entropy of the tree based the node degrees distribution

0 3

S 2

3

2

6 7 8

4

nH ∗ (T ) ≤ 2n

Jansson, Sadakane and Sung (Soda07)

An example: ultrasuccinct encodings of trees

H

(T ) :=

log

i

1

5

0 0 2 0 0

where ni is the number of nodes with i children

nH

(T ) + o(n) bits

(entropy of a tree)

with a given node degrees distribution

Encoding ”geometric” data

”regular” meshes

random graphs

Geometric object



 x y z





between 30 et 96 bits/vertex

Geometric information

vertex triangle

1 reference to a triangle 3 references to vertices 3 references to triangles

log n ou 32 bits

Connectivity information

Combinatorial object

connectivity

2 × n × 6 × log n n × 1 × log n

13n log n

416n bits

Enumeration of planar triangulations (Tutte, 1962)

Enumeration and (Tutte’s) entropy of planar triangulations

Ψ_n = 2(4n + 1)!

(3n + 2)!(n + 1)! ≈ 16 27

r 3

2π n^−5/2(256 27 )ⁿ

1

n log₂ Ψ_n ≈ log₂(256

27 ) ≈ 3.2451 bits/vertex Entropy

Graph encodings: trees decompositions

Edgebreaker

CCCRCCRCCRECRRELCRE

C CC

R C

R C

C

C R

S C

R R L E

C E R

Multiple parenthesis encoding

( [ [ [ ) ( ] ( ] ( ] [ [ [ ) [ ) ( ] ] [ ) . . .

V₅V₅V₆V₅V₄V₅V₈V₅V₅V₄S₄V₃V₄ T ouma Gotsman(⁰98)

1101000110000010010000011001000000000

P oulalhon Schaef f er(2003)

General visual framework (Isenburg Snoeyink)

T ouma Gotsman(⁰98)

Edgebreaker via Schnyder woods (or canonical orderings)

Mesh compression Graph encoding Succinct representations

Jacobson (Focs89)

Munro and Raman (Focs97) Chuang et al. (Icalp98)

Chiang et al. (Soda01)

C-A, Devillers and Schaeffer (SoCG06)

C-A, Devillers and Schaeffer (Wads05, CCCG05)

Barbay et al. (Isaac07) Nakano et al. (2008) Poulalhon Schaeffer

(Icalp03)

Fusy et al. (Soda05)

Blandford Blelloch (Soda03) C-A, Fusy Lewiner

(SoCG08) Turan (’84)

Keeler Westbrook (’95) Computer graphics Graph theory / combinatorics

He et al. (’99) Edgebreaker

V alence (degree) Rossignac (’99)

Touma and Gotsman (’98) Alliez and Debrun

algorithms and DS

Cut − border machine Isenburg

Khodakovsky

Gumhold et al.

(Siggraph ’98)

Gumhold (Soda ’05)

3.2451n 3.67n

3.28n

Lope et al. (’03) Lewiner et al. (’04)

. . . . (many many others)

. . . . (many others)

≈ 2n ? n

Chuang et al. (Icalp98)

4n

≈ 2.5n

4n

3.2451n

Graph encoding: the combinatorial approach

Entropy of triangulations =Tutte’s entropy (from enumeration)

E(T) := 1

n log2 Ψn ≈ log2(256 27

) ≈ 3.2451 bits/vertex

Graph encoding: the combinatorial approach

Idea (combinatoricians): find a good description of planar graphs

Schnyder woods (also known as Schnyder trees or realizers)

(Schnyder ’90)

A partition T₀, T₁, T₂ of the internal edges of T s.t. :

i) edge are colored and oriented in such a way that each inner nodes has exaclty one outgoing edge of each color

ii) colors and orientations around each inner node must respect the local Schnyder condition

x

x

x

n nodes

Graph encoding: the combinatorial approach

Important properties of Schnyder woods (with applications)

Applications

• graph enumeration, graph drawing, graph encoding, ...

Theorem

A graph G is planar if and only if its dimension is at most 3 (Schnyder, Felsner, Trotter)

Theorem

A graph G is planar if and only if its dimension is at most 3

Theorem

The three set T₀, T₁, T₂ are spanning trees of (the inner nodes of) T:

x1

x₃ x₂

R₁(v) v R₃(v)

R2(v) x₁

x₃ x2

α₁ v α3 α₂

Combinatorial interpretation of barycentric embedding

Applications: graph compression and succinct encoding

0 2 3 1

5 4 6 8 7

9 10

11

4 32 0 5

6 7

8 9 10

1 0

1 3 2

5 4

6 7 8

9 )

(

( [[ [[ ( ( (] ] ]{[ [ ) )({ }]{)(}]{[ [ [ [) )({ }]{)(}] ]( ) )({ { }}]{)(}}}])) S

T

T

T

Succinct encoding (Chuang-Garg-He-Kao-Lu Icalp ’98)

(Barbay-Castelli Aleardi-He-Munro Isaac’07) Theorem

The three set T₀, T₁, T₂ are spanning trees of (the inner nodes of) T:

(Nakano et al. 2008)

Th´eor`eme. (Poulalhon–Schaeffer Icalp 03)

Bijection between plane trees of size n, having two stems per node, and the class of rooted planar triangulations with n + 2 vertices.

Theorem. (Tutte 62) The number of planar triangulations with n + 2 vertices is

2(4n−3)!

(3n−1)!n! (²⁵⁶₂₇ )ⁿ .

a new nice interpretation of Tutte’s formula:

|T_n| = _2n² · |A⁽²⁾_n |.

Optimal coding and sampling (planar case)

?

genus g triangulations

Bijective enumeration, optimal coding and sampling

planar triangulations Schnyder (Soda ’90)

Poulalhon and Schaeffer (Icalp ’03)

Castelli-Aleardi, Fusy and Lewiner (SoCG08)

work in progress

Optimal encoding is not adaptive

achieving ”bad” compression ratios for easy istances

Is Tutte’s entropy really always a good measure?

Near optimal encoding

Entropy of triangulations = entropy of vertex degrees sequence

E (T ) :=

∞

X

i=3

p

log 1 p

”regular” meshes

node degree distribution: low dispersion around average degree (6)

(from Surface Reconstruction or Geometric Modeling)

adaptive behaviour

adaptive behaviour

Graph encoding: an heuristic approach

Entropy of triangulations = entropy of vertex degrees sequence

E (T ) :=

∞

X

i=3

p

log 1 p

E (T ) ≤ log

≈ 3.2451 . . .= Tutte entropy !

quite surprisingly

∞

X

i=3

p_i = 1

∞

X

i=3

i · p_i = 6 Solve an optimization problem

using Lagrange multipliers

Alliez Desbrun (Eurographics ’01)

adaptive behaviour

How to define an interesting measure of disorder for graphs?

but not optimal encoder

E (T ) < log

≈ 3.2451 . . .= Tutte entropy !

unfortunately

Gotsman (2003)

E (T ) ≈ 3.2364 . . .

(via analytic calculations) p_i ≈ 4(πi)⁻¹² (3

4)ⁱ

Degree driven approach

Can we design a (non trivial) optimal encoder with adaptive behaviour?

Open intereting questions