Repr´ esentations succinctes optimales pour les maillages

Repr´ esentations succinctes optimales pour les maillages

Luca Castelli Aleardi

(en collaboration avec Olivier Devillers et Gilles Schaeffer)

19 octobre 2006, ENST Paris

Projet Geometrica LIX

INRIA Sophia-Antipolis Ecole Polytechnique

Succinct and compact representations

Given a class ^Cm of objects of size ^m, the goal is to design a space efficient data structure such that:

• queries on objects are answered in constant time;

• the encoding is succinct: the cost of an object ^R ∈ Cm

matches asymptotically the entropy of the class size(R) = log₂ kCmk(1 + o(1))

• or compact: we content of a cost

size(R) = O(log₂ kCmk)

• for dynamic data structures: updates are supported in O(lg^c m) amortized time

Compact representations: an example

Rooted trees with n vertices

enumeration of binary trees with n vertices:

kBⁿk = 1 n + 1

2n

n

≈ 2²ⁿn⁻³² (1)

Example

Optimal (implicit) encoding for compression

• size: log2 kB_nk = 2n + O(lg n) bits

• no efficient navigation

11101010001101001100

Example: explicit representation

Pointers-based representation

• size: 2n lg n bits

• constant time navigation between nodes

Clef t Cright

/

Cright

parent

Clef tCright

parent

/ /

parent

/ /

parent Clef tCright

parent

/ /

parent

/ /

parent /

lgn

lgn lgn

lgn

lgn

Compact and succinct representations

(Jacobson Focs89, Munro et Raman Focs97)

• size: O(n) bits - adjacency queries in O(lg n) time

• size: 2n + o(n) bits - adjacency queries in O(1) time

01100 01000 11010

11101

lgn

Motivation

Combinatorial information describing incidence relations

Which information?

Connectivity

Motivation

Geometry information (vertex coordinates)

Geometric object

Geometric information











x y z











between 30 and 96 bits (192 bits ?)

Motivation

Usual triangle-based mesh representation

Geometric object

Connectivity information Geometric information

vertex:

triangle:

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

logn to 32 bits

Motivation

Mesh compression algorithms

Touma Gotsman Edgebreaker [Rossignac]

Poulhalon Schaeffer General underlying idea

Encoding strategies based on a local (global) conquest

Graph encoding

(Similarities between spanning-tree based methods, Isenburg and Snoeyink)

Edgebreaker

CCCRCCRCCRECRRELCRE

C CC

R C

C R

C C R

S C

R R L E

C E R

V₅V₅V₆V₅V₄V₅V₈V₅V₅V₄S₄V₃V₄

T ouma Gotsman(^′98)

Canonical orderings and multiple parentheses Chuang et al. (Icalp 98) - Chiang et al. (Soda 01)

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

1101000110000010010000011001000000000

P oulalhon Schaef f er(2003)

Motivation

Mesh compression algorithms

Mesh

comp ress ion

Compact representation

VRML, 288 or 114 bits/vertex

[Touma Gotsman] 2bits/vertex (near-optimal)

Pointer based representation: 208 bits/triangle 2.175 bits/triangle

[Poulalhon Schaeffer] 3.24bits/vertex (optimal)

Tutte’s entropy (triangulations)

(information theory asymptotic lower bound)

enumeration of rooted planar triangulations on n vertices:

Ψ_n = 2(4n + 1)!

(3n + 2)!(n + 1)! ≈ 16 27

r 3

2πn^−5/2(256 27 )ⁿ Tutte’s entropy (1962):

e = 1

n log2 Ψ_n ≈ log2(256

27 ) ≈ 3.2451 bits/vertex

Planar Triangulations with a boundary

n + 1 internal vertices, m = 2n + k faces f(n, k) = 2 · (2k − 3)! (2k + 4n − 1)!

(k − 1)! (k − 3)! (n + 1)! (2k + 3n)!

f^′(m, k) = 2 · (2k − 3)! (2m − 1)!

(k − 1)! (k − 3)! (^m₂^−k + 1)!

counting planar triangulations with m faces F(m) = lg(

m

X

k≥3

f^′(m, k)) ≈ 2.175m

3.24 bits/vertex = 1.62 bits/face < 2.17 bits/face

Triangulations with a boundary

(Castelli Aleardi, Devillers and Schaeffer, WADS 2005) Theorem. For planar triangulations with a boundary having ^m faces, there exists an optimal succinct

representation supporting efficient navigation in ^O(1) time, requiring

2.175m + O(mlg lg m

lg m ) = 2.175m + o(m) bits

For triangulations of genus ^g surfaces (^g = o(_lg^m_m)) the same representation requires

2.175m + 36(g − 1) lg m + O(mlg lg m

lg m + g lg lg m) bits

Comparison: space efficiency

graphs with n vertices, e edges, m faces, genus g

Encoding queries 3-conn. triangl. g > 0 dynamic Jacobson (Focs89) O(lg n) 64n 64n no

Munro Raman (Focs97) O(1) 8n + 2e 7m no Chuang et al. (Icalp98) O(1) 2e + 2n 3.5m no Chiang et al. (Soda01) O(1) 2e + 2n 4m no Blandford et al. (Soda03) O(1) O(n) O(m) O(n) Castelli et al. (Wads05

Cccg05)

O(1) no 2.175m 2.175m +O(g lg m Castelli et al. (SoCG06) O(1) 2e 1.62m no

Literary digression ”La le¸con”, (Eug`ene Ionesco, 1951)

Au cours d’une le¸con priv´ee en pr´eparation du doctorat total, une jeune fille discute d’arithm´etique avec son professeur.

(le professeur)Ecoutez-moi, Mademoiselle, si vous n’arrivez pas `a´ comprendre profond´ement ces principes, ces arch´etypes

arithm´etiques, vous n’arriverez jamais `a faire correctement un travail de polytechnicien... combien font, par exemple,

3.755.918.261 multipli´e par 5.162.303.508?

(l’´el`eve, tr`es vite) Ca fait 193891900145...¸

(le professeur, de plus en plus ´etonn´e calcule mentalement)Oui... Vous avez

raison... le produit est bien... (il bredouille inintelligiblement)...

Mais comment le savez-vous, si vous ne connaissez pas les principes du raisonnement arithm´etique?

(l’´el`eve) C’est simple. Ne pouvant me fier `a mon raisonnement, j’ai appris par coeur tous les r´esultats possibles de toutes les multiplications possibles.

Decomposing T into sub-triangulations

• compute small triangulations of size bewteen ¹₃ lg² m and lg² m;

• decompose into tiny triangulations of size between

1

12 lg m and ¹₄ lg m.

Initial triangulation with m triangles Tiny triangulations with Θ(lgm) triangles Small triangulations with Θ(lg²m) triangles

Decomposing T into sub-triangulations

• we compute small triangulations having between ¹₃ lg² m and lg² m triangles;

• we decompose small triangulations into tiny

triangulations containing between ₁₂¹ lg m and ¹₄ lg m triangles.

Graph G linking tiny triangulations having Θ(_lg^m_m) nodes

A sub-graph Gi linking Θ(lgm) tiny triangulations lying in the same small

Graph F linking small triangulations with Θ(_lg^m2m) nodes

Overview: representation of a small triangulation

• adjacency relations are described by map ^Gi;

• internal connectivity is implicitly represented (variable size pointers)

• boundary neighboring relations are represented by boundary coloring (variable length bit-vector)

pointer to tiny triangulation of size t

pointer to rank-select of size b and weight s t, b, s

split small into tiny

first vertex

0

1 0

0 1 1

1

1 0

0

0 0

0

Graph G

linking adjacent tiny triangulations

• Gi has a node for each tiny triangulation and an arc for each pair of adjacent tiny triangulations;

• Gⁱ is a planar map, having faces of degree at least 3, multiple edges and loops are allowed;

i

adjacency relations between tiny triangulations

• Because of Euler’s formula, the overall number of arcs in maps ^Gi is:

X

i

kE(Gi)k = O( m lg m)

Dominant cost analysis

The Catalog A_r contains all triangulations with r triangles A reference to an element in A_r costs 2.17r bits

k A k = 2

r

Total space used

• Catalog of all different tiny triangulations

O(m^412.17 lg² m lg lg m) = o(m)

• catalog of bit-vectors (with Rank/Select)

O(m^412.17 lg m lg lg m) = o(m)

• representation of graph ^F: ^O( ^m

lg² m lg m) = o(m)

• graphs ^Gi

2.17m + O(mlg lg m lg m )

Optimality

(SoCG 2006)

Proposition (Castelli Aleardi, Devillers and Schaeffer).

For planar maps (triangulations and 3-connected graphs) it is possible to design two optimal succinct representations, supporting efficient navigation in ^O(1) time, whose storage requirements match asymptotically the entropy of the

respective classes. More precisely, our representations use:

• 1.62m bits, for triangulations (without boundary) with m faces;

• 2e bits, for 3-connected graphs with e edges.

Optimal encodings

(Poulalhon/Schaeffer 03, Fusy/Poulalhon/Schaeffer 05)

Original local closures

New ”color-based” closures

”color-based” closure for triangulations

11010001010000100000000000 00100100100001

n nodes, 2n stems

n nodes, 2n stems n nodes, 2n stems

n nodes, 2n stems

n nodes, 2n stems n nodes, 2n stems n nodes, 2n stems n nodes, 2n stems

New splitting strategies

carte G

New splitting strategies

Dominant cost analysis

The Catalog A_r contains all triangulations with r triangles A reference to an element in A_r costs 2.17r bits

k A k = 2

r

Dominant cost analysis

Catalog A_r,w: all trees with r nodes, 2r stems and w false stems A reference to an element in A_r,w costs 2.17r + w lg r bits

kA_r,wk = 2^3.24r · ²_w^r

3.24r + w lg r bits index

Future and related works

• Design of a practical efficient implementation (joint work with A. Mebarki and O. Devillers, CCCG ’06)

• extension: realizers, canonical orderings and ”spanning tree” encodings for higher genus surfaces

Graph encodings

(higher genus triangulations)

Edgebreaker

CCCRCCRCCRECRRELCRE

C CC

R C

C R

C C R

S C

R R L E

C E R

V₅V₅V₆V₅V₄V₅V₈V₅V₅V₄S₄V₃V₄

T ouma Gotsman(^′98)

Edgebreaker on a torus

Canonical orderings and multiple parentheses Chuang et al. (Icalp 98) - Chiang et al. (Soda 01)

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

1101000110000010010000011001000000000

P oulalhon Schaef f er(2003)

?

Repr´esentations succinctes optimales pour les maillages – p.35/35