Luca Castelli Aleardi

21 april 2009, Dagstuhl

Graph encoding: an overviewalgorithmics and combinatorics of (planar) graphs

LIX Ecole Polytechnique

Domain and problems

an overview

Triangulations and graphs

Geometric data

GIS Technology

ApplicationsSurface recontruction

Very large geometric data

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

186 millions vertices6 Giga bytes (for storing on disk)

tens of minutes (for loading the model fromdisk)

David statue (Stanford’s Digital Michelan-gelo Project, 2000)2 billions polygons32 Giga bytes (without compression)

No existing algorithm nor datastructure for dealing with the entiremodel

Research topicsMesh 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 01 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

‖Bn‖ = 1n+1

(2nn

)≈ 22nn−

32

Enumeration of plane trees with n edges

An example: plane trees

Enumeration and entropy of plane trees

Optimal encoding matching asymptotically theinformation-theory lower bound

log2‖Bn‖ = 2n + O(lg n)

1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 01 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

( ( ( ( ) ) )

11000 000010000100001 00

b1 b2 b3 b4 b5

5 2 4 4 5

0 3 2 3 2D

B

T

32

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 thetree based the node degrees distribution

032S

3

2

6 874

nH ∗ (T ) ≤ 2n

Jansson, Sadakane and Sung (Soda07)

An example: ultrasuccinct encodings of trees

H∗(T ) :=∑

ini

nlog n

ni

1

50 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” meshesrandom graphs

Geometric object

xyz

between 30 et 96 bits/vertex

Geometric information

vertex

triangle

1 reference to a triangle

3 references to vertices3 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

Ψn =2(4n + 1)!

(3n + 2)!(n + 1)!≈

16

27

√3

2πn−5/2(

256

27)n

1

nlog2 Ψn ≈ log2(

256

27) ≈ 3.2451 bits/vertex

Entropy

Graph encodings: trees decompositions

Edgebreaker

CCCRCCRCCRECRRELCREC C

C

R

C

CR

CC

RS

C

R REL

CRE

Multiple parenthesis encoding

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

V5V5V6V5V4V5V8V5V5V4S4V3V4

Touma Gotsman(′98)

1101000110000010010000011001000000000

Poulalhon Schaeffer(2003)

General visual framework (Isenburg Snoeyink)

Touma Gotsman(′98)

Edgebreakervia 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

IsenburgKhodakovsky

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

E(T ) :=1

nlog2 Ψn ≈ log2(

256

27) ≈ 3.2451 bits/vertex

Graph encoding: the combinatorial approachIdea (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 sucha way that each inner nodes has exacltyone outgoing edge of each color

ii) colors and orientations aroundeach inner node must respect thelocal Schnyder condition

x0 x1

xn−1

n nodes

Graph encoding: the combinatorial approachImportant properties of Schnyder woods (with applications)

Applications

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

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

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

TheoremThe three set T0, T1, T2 are spanningtrees of (the inner nodes of) T :

x1

x3x2

R1(v)v

R3(v)R2(v)

x1

x3x2

α1

vα3 α2

Combinatorial interpretationof barycentric embedding

Applications: graph compression and succinct encoding

0 12 345

6

78910

11

04 32

5

6

78

910

1 0

12

34

56

7

89

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

T0 T1

T2

Succinct encoding (Chuang-Garg-He-Kao-Lu Icalp ’98)(Barbay-Castelli Aleardi-He-Munro Isaac’07)

TheoremThe three set T0, T1, T2 are spanningtrees of (the inner nodes of) T :

(Nakano et al. 2008)

Theoreme. (Poulalhon–Schaeffer Icalp 03)Bijection between plane trees of size n, having two stems per node, andthe class of rooted planar triangulations with n + 2 vertices.

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

2(4n−3)!(3n−1)!n!

� ( 25627

)n .

a new nice interpretation of Tutte’s formula:

|Tn| = 22n · |A(2)

n |.

Optimal coding and sampling (planar case)

?

genus g triangulationsBijective 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 adaptiveachieving ”bad” compression ratios for easy istancesIs Tutte’s entropy really always a good measure?

Near optimal encodingEntropy of triangulations = entropy of vertex degrees sequence

E(T ) :=∞∑i=3

pi log1

pi

”regular” meshes

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

(from Surface Reconstruction or Geometric Modeling)

adaptive behaviour

adaptive behaviourGraph encoding: an heuristic approach

Entropy of triangulations = entropy of vertex degrees sequence

E(T ) :=∞∑i=3

pi log1

pi

E(T ) ≤ log225627

≈ 3.2451 . . .= Tutte entropy !

quite surprisingly

∞∑i=3

pi = 1

∞∑i=3

i · pi = 6

Solve an optimization problemusing Lagrange multipliers

Alliez Desbrun (Eurographics ’01)

adaptive behaviour

How to define an interesting measure of disorder for graphs?

but not optimal encoder

E(T ) < log225627

≈ 3.2451 . . .= Tutte entropy !

unfortunately

Gotsman (2003)

E(T ) ≈ 3.2364 . . .

(via analytic calculations)

pi ≈ 4(πi)−12 (

3

4)i

Degree driven approach

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

Open intereting questions