Luca Castelli Aleardi

Optimal succinct representations of planar maps

Luca Castelli Aleardi

(joint work with Olivier Devillers and Gilles Schaeffer)

6th june 2006, SoCG 2006

Projet Geometrica LIX

INRIA Sophia-Antipolis Ecole Polytechnique

Succinct and compact representations

Given a class ^{C m} 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 ∈ C m

matches asymptotically the entropy of the class size ( R ) = log ₂ k C m k(1 + o (1))

• or compact : we content of a cost

size ( R ) = O (k C m k)

• 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:

kB ⁿ k = 1 n + 1

2 n

n

≈ 2 ^{2 n} n ⁻

(1)

Example

Optimal (implicit) encoding for compression

• size: log 2 kB _n k = 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

C

C

/

C

parent

C

C

parent

/ /

parent

/ /

parent C

C

parent

/ /

parent

/ /

parent /

lg n

lg n lg n

lg n

lg n

Compact and succinct representations

(Jacobson Focs89, Munro et Raman Focs97)

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

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

01100 01000 11010

11101

lg n

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

log n 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(4 n + 1)!

(3n + 2)!(n + 1)! ≈ 16 27

r 3

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

e = 1

n log 2 Ψ _n ≈ log 2 ( 256

27 ) ≈ 3 . 2451 bits/vertex

Planar Triangulations with a boundary

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

( k − 1)! ( k − 3)! ( n + 1)! (2 k + 3 n )!

f ^′ (m, k) = 2 · (2k − 3)! (2m − 1)!

(k − 1)! (k − 3)! ( ^m ₂ ^{− k} + 1)!

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

m

X

k ≥ 3

f ^′ ( m, k )) ≈ 2 . 175 m

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 . 175 m + O ( m lg lg m

lg m ) = 2 . 175 m + 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(m lg 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 ) no

Munro Raman (Focs97) O(1) 8n + 2e 7m no Chuang et al. (Icalp98) O (1) 2 e + 2 n 3 . 5 m no Chiang et al. (Soda01) O (1) 2 e + 2 n 4 m no Blandford et al. (Soda03) O (1) O ( n ) O ( m ) O ( n ) Castelli Devillers Scha-

effer(Wads Cccg05)

O (1) no 2 . 175 m 2 . 175 m

our new encodings O (1) 2 e 1 . 62 m no

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

During a private lesson, a very young student, preparing herself for the total doctorate, talks about arithmetics with her teacher.

(teacher) If you cannot deeply understand these principles,

these arithmetic archetypes, you will never perform correctly a

”polytechnicien” job. For example, what is 3 . 755 . 918 . 261 multiplied by 5 . 162 . 303 . 508?

(student, very quickly) The result is 193891900145...

(teacher, very astonished): How have you computed it, if you do not know the principles of arithmetic reasoning?

(student) Simple: I have learned by heart all possible results of

all possible multiplications.

Decomposing T into sub-triangulations

• compute small triangulations of size bewteen ¹ ₃ lg ² m and lg ² m ;

• decompose into tiny triangulations of size between

1

12 lg m and ¹ ₄ 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 ¹ ₃ lg ² m and lg ² m triangles;

• we decompose small triangulations into tiny

triangulations containing between ₁₂ ¹ lg m and ¹ ₄ 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 ^{G i} ;

• 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 _i linking adjacent tiny triangulations

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

• G ⁱ 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 ^{G i} is:

X

i

k E ( G i )k = O ( m

lg m )

Total space used

• Catalog of all different tiny triangulations

O(m

^{2 . 17} lg ² m lg lg m) = o(m)

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

O ( m

^{2 . 17} lg m lg lg m ) = o ( m )

• representation of graph ^F : ^O ( ^m

lg

m lg m ) = o ( m )

• graphs ^{G i}

2.17m + O(m lg lg m

lg m )

Optimality

Our contribution

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 . 62 m bits, for triangulations (without boundary) with m faces;

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

Optimal encodings

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

Original local closures

New ”color-based” closures

11010001010000100000000000 00100100100001 nnodes, 2nstems

nnodes, 2nstems nnodes, 2nstems

nnodes, 2nstems nnodes, 2nstems nnodes, 2nstems nnodes, 2nstems

”color-based” closure for 3 -connected graphs

”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

Dominant cost analysis

The Catalog A _r contains all triangulations with r triangles A reference to an element in A _r costs 2 . 17 r bits

k A k = 2 ^2.17r

r _2.17r bits reference

Dominant cost analysis

Catalog A _r,w : all trees with r nodes, 2 r stems and w false stems A reference to an element in A _r,w costs 2 . 17 r + w lg r bits

k A _r,w k = 2 ^{3 . 24 r} ·

² _w ^r

3.24r + w lg r bits index

Future and related works

• Design of a practical efficient implementation (with A.

Mebarki and O. Devillers)

• 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)

?