[PPT] - Schnyder woods: from graph encoding to graph drawing v 2 INF562, 9 f PowerPoint Presentation

SLIDE 1

Luca Castelli Aleardi

INF562, 9 f´ evrier 2016

v0 v1 v2

Schnyder woods: from graph encoding to graph drawing

SLIDE 2

Some facts about planar graphs

(”As I have known them”)

SLIDE 3

G planar if and only if G contains neither K5 nor K3,3 as minors Thm (Schnyder, Trotter, Felsner) Every planar graph with n vertices is isomorphic to the intersection graph of n disks in the plane. Thm (Koebe-Andreev-Thurston)

Some facts about planar graphs

Thm (Kuratowski, excluded minors) G planar if and only if dim(G) ≤ 3 Thm (Y. Colin de Verdi` ere) G planar if and only if µ(G) ≤ 3 (µ(G) = multiplicity of λ2 of a generalized laplacian) vN S2 vN M 4 −1 . . . −1 5 . . . 3 LG = . . . . . . . . . . . . . . . . . .

LG[i, k] = { −AG[i, j] deg(vi)

SLIDE 4

Every planar graph with n vertices is isomorphic to the intersection graph of n disks in the plane. Thm (Koebe-Andreev-Thurston)

Using circles to measure distances

(images by N. Bonichon)

SLIDE 5

Every planar graph with n vertices is isomorphic to the intersection graph of n disks in the plane. Thm (Koebe-Andreev-Thurston)

Using circles to measure distances

(images by N. Bonichon) (images by R. Silveira) Not every planar triangulation is Delaunay realizable

SLIDE 6

TD-Delaunay: triangular distance Delaunay triangulations Thm (de Fraysseix, Ossona de Mendez, Rosenstiehl, ’94)

Using triangles to measure distances

(images by S. Felsner)

Chew, ’89

(images by N. Bonichon)

Every planar triangulation is TD-Delaunay realizable

SLIDE 7

Planar triangulations

1 2 3 4 5 10 8 12 21 19 24 15 22 17 23 18 16 7 13 9 11 20 6 14

φ = (1, 2, 3, 4)(17, 23, 18, 22)(5, 10, 8, 12)(21, 19, 24, 15) . . . α = (2, 18)(4, 7)(12, 13)(9, 15)(14, 16)(10, 23) . . .

n − e + f = 2 e = 3n − 6

S2

SLIDE 8

Schnyder woods and canonical orderings: overview of applications

(graph drawing, graph encoding, succinct representations, compact data structures, exhaustive graph enumeration, bijective counting, greedy drawings, spanners, contact representations, planarity testing, untangling of planar graphs, Steinitz representations of polyhedra, . . .)

SLIDE 9

Some (classical) applications

bijective counting, random generation

1 2 3 4 5 6 7 8 9 10 11 4 3 2 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9

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

[ [ [ [

] ] ]

]

{ } { } { { } { } { { { } } } { } } S

T0 T1 T2

(Poulalhon-Schaeffer, Icalp 03)

Graph encoding

Thm (Schnyder ’90) planar straight-line grid drawing (on a O(n × n) grid)

cn =

2(4n+1)! (3n+2)!(n+1)!

⇒ optimal encoding ≈ 3.24 bits/vertex (Chuang, Garg, He, Kao, Lu, Icalp’98) (He, Kao, Lu, 1999)

SLIDE 10

More (”recent”) applications

u v a b

Every planar triangulation admits a greedy drawing (Dhandapani, Soda08) (conjectured by Papadimitriou and Ratajczak for 3-connected planar graphs)

Greedy routing

Schnyder woods, TD-Delaunay graphs, orthogonal surfaces and Half-Θ6-graphs

[ Bonichon et al., WG’10, Icalp ’10, ...]

SLIDE 11

Schnyder woods

(the definition)

SLIDE 12

Schnyder woods: (planar) definition

v0 v1 n nodes v2

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

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

A Schnyder wood of a (rooted) planar triangulation is partition of all inner edges into three sets T0, T1 and T2 such that

[Schnyder ’90] rooted triangulation on

SLIDE 13

Schnyder woods: equivalent formulation

v0 v1 v2 [3-orientation] [normal labeling]

0 0 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

1 1 2 3-connected graphs [Felsner]

1 2

SLIDE 14

Schnyder woods: spanning property

v0 v1 v2 [Schnyder ’90]

Theorem The three sets T0, T1, T2 are spanning trees of the inner vertices of T (each rooted at vertex vi)

v1 v0

T2 T1

v0 v1 [incremental vertex shelling, Brehm’s thesis]

Theorem Every planar triangulation admits a Schnyder wood, which can be computed in linear time.

The traversal starts from the root face

Invariant:

vk u

SLIDE 21

Schnyder woods: existence (algorithm I)

v0 v1 [incremental vertex shelling, Brehm’s thesis]

Theorem Every planar triangulation admits a Schnyder wood, which can be computed in linear time.

The traversal starts from the root face

Invariant:

vk u

SLIDE 22

Schnyder woods: existence (algorithm I)

v0 v1 [incremental vertex shelling, Brehm’s thesis]

Theorem Every planar triangulation admits a Schnyder wood, which can be computed in linear time.

The traversal starts from the root face

Invariant:

SLIDE 32

Barycentric representation of a planar graph

c b a v2 v v1 v0

v = v0a + v1b + v2c where vi is the normalized area

Barycentric representation def:

f(v) − → (v0, v1, v2) ∈ R3 v0 + v1 + v2 = 1 , for each vertex v for each edge (x, y) ∈ E and each vertex z / ∈ {x, y} there is an index k ∈ {0, 1, 2} such that xk < zk yk < zk

SLIDE 33

Barycentric representation of a planar graph

Barycentric representation def:

f(v) − → (v0, v1, v2) ∈ R3 v0 + v1 + v2 = 1 , for each vertex v

(1, 0, 0) (0, 1, 0) (0, 0, 1)

a b c

SLIDE 34

Barycentric representation of a planar graph

Barycentric representation def:

f(v) − → (v0, v1, v2) ∈ R3 v0 + v1 + v2 = 1 , for each vertex v for each edge (x, y) ∈ E and each vertex z / ∈ {x, y} there is an index k ∈ {0, 1, 2} such that xk < zk yk < zk

(1, 0, 0) (0, 1, 0) (0, 0, 1) (1, 0, 0) (0, 1, 0) (0, 0, 1) f(x) f(y) f(z)

no vertex z in the gray triangle defined by f(x), f(y)

max(x2, y2) max(x0, y0)

SLIDE 35

Barycentric representation of a planar graph

Theorem A barycentric representation defines a planar straight-line drawing of G, in the plane spanned by (1, 0, 0), (0, 1, 0) and (0, 0, 1).

for each edge (x, y) ∈ E and each vertex z / ∈ {x, y}, f(z) cannot lie on (f(x), f(y))

(1, 0, 0) (0, 1, 0) (0, 0, 1) f(x) f(y) f(z) max(x2, y2) max(x0, y0)

f(z) = tf(x) + (1 − t)f(y) , for t ∈ [0, 1] zk = txk + (1 − t)yk) < tzk + (1 − t)zk = zk for k ∈ {0, 1, 2}

SLIDE 36

Barycentric representation of a planar graph

Theorem A barycentric representation defines a planar straight-line drawing of G, in the plane spanned by (1, 0, 0), (0, 1, 0) and (0, 0, 1).

given two edges (x, y), (u, v) ∈ E they cannot cross

f(x) f(y) f(u)

ui, vi < xi

f(v)

by definition there are four indices i, j, k, l ∈ {0, 1, 2} uj, vj < yj xk, yk < uk xl, yl < vl i = k, l and j = k, l i = j or k = l assume i = j = 2 there exists a separating line l parallel to [(1, 0, 0), (0, 1, 0)]

l

if i = k we would have uk < xk vk < xk contradicting xk, yk < uk

SLIDE 37

Face counting algorithm

(Schnyder algorithm, 1990)

SLIDE 38

Face counting algorithm

x1 x3 x2 R1(v) v R3(v) R2(v)

v = |R1(v)|

|T |

x1 + |R2(v)|

|T |

x2 + |R3(v)|

|T |

x3 where |Ri(v)| is the number of triangles

Theorem (Schnyder, Soda ’90) For a triangulation T having n vertices, we can draw it on a grid of size (2n − 5) × (2n − 5), by setting x1 = (2n − 5, 0), x2 = (0, 0) and x3 = (0, 2n − 5).

x1 x3 x2 α1 v α3 α2

v = α1x1 + α2x2 + α3x3 where αi is the normalized area

Geometric interpretation

SLIDE 39

Non crossing paths, face partition

c b a R2(v) v R1(v) R0(v) c b a v

SLIDE 40

Non crossing paths, face partition

c b a

Ri(v)

v u c b a

Ri(u)

v u

if u ∈ Ri(v) Ri(u) ⊂ Ri(v) ui < vi

SLIDE 41

Linear-time implementation

c b a

Ri(v)

v u c b a v

Pi+1(v) Pi−1(v) how to efficiently compute |Ri(v)|? t1 t2 t3 t4

SLIDE 42

Face counting algorithm

⇒

T endowed with a Schnyder wood Input: T

a b c d e f g h i d a b c d e f g h i

→ (0, 0) → (0, 1) → (1, 0) → ( 5

13, 6 13)

→ ( 9

13, 1 13)

→ ( 7

13, 4 13)

→ ( 3

13, 3 13)

→ ( 4

13, 8 13)

→ ( 1

13, 4 13)

SLIDE 43

Face counting algorithm: proof (sketch)

⇒

T endowed with a Schnyder wood Input: T

a b c d e f g h i d a b c d e f g h i

→ (13, 0, 0) → (0, 13, 0) → (0, 0, 13) → ( → (9, 3, 1) → (2, 7, 4) → (7, 3, 3) → (1, 4, 8) → (8, 1, 4)

d

5, 6, 2 )

SLIDE 44

Face counting algorithm: proof (sketch)

⇒

T endowed with a Schnyder wood Input: T

a b c d e f g h i a b c d e f g h i

→ (13, 0, 0) → (0, 13, 0) → (0, 0, 13) → ( → (9, 3, 1) → (2, 7, 4) → (7, 3, 3) → (1, 4, 8) → (8, 1, 4)

d

5, 6, 2 )

SLIDE 45

Face counting algorithm: proof (sketch)

⇒

T endowed with a Schnyder wood Input: T

a b c d e f g h i a b c d e f g h i

→ (13, 0, 0) → (0, 13, 0) → (0, 0, 13) → ( → (9, 3, 1) → (2, 7, 4) → (7, 3, 3) → (1, 4, 8) → (8, 1, 4)

d

5, 6, 2 )

SLIDE 46

Graph encoding

SLIDE 47

(practical) motivation

Geometric v.s combinatorial information

Geometry between 30 et 96 bits/vertex vertex triangle 1 reference to a triangle 3 references to vertices 3 references to triangles ”Connectivity”: the underlying triangulation

13n log n 416n bits

vertex coordinates adjacency relations between triangles, vertices

r

#{triangulations} =

2(4n + 1)! (3n + 2)!(n + 1)! ≈ 16 27

3

2πn−5/2 256 27 n

⇒ entropy = log2

256 27 ≈ 3.24 bpv.

David statue (Stanford’s Digital Michelangelo Project, 2000) 2 billions polygons

32 Giga bytes (without compression)

No existing algorithm nor data structure for dealing with the entire model

SLIDE 48

A simple encoding scheme

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

2 3 4 5 6 7 8 9 10 11

[[[[ ] ] ][[[ [ ]][ ]][[[[[ [ ]][ ]] ][ [ ]]][ ]]]] G \ T

T T

([[[)(](][[[)])(]][) . . . S(G)

Turan encoding of planar map (1984) length(S) = 2e symbols (2 log2 4)e = 4e = 12n bits G = (V, E) |V | = n |E| = e

T := (any) vertex spanning tree of G

parenthesis word of size 2n

12n bits encoding scheme

parenthesis word of size 2n

1

SLIDE 49

A more efficient encoding

Canonical orderings - Schnyder woods (He, Kao, Lu ’99) 2 3 4 5 6 7 8 9 10 11 T0 T1 T2

SLIDE 50

A more efficient encoding

Canonical orderings - Schnyder woods (He, Kao, Lu ’99) 2 3 4 5 6 7 8 9 10 11

T1 T2

T 0

1

SLIDE 51

A more efficient encoding

Canonical orderings - Schnyder woods (He, Kao, Lu ’99) 2 3 4 5 6 7 8 9 10 11

T2

T1 is redundant: reconstruct from T0, T2

1 1

T 0

SLIDE 52

A more efficient encoding

Canonical orderings - Schnyder woods (He, Kao, Lu ’99) 2 3 4 5 6 7 8 9 10 11

T2

T1 is redundant: reconstruct from T0, T2

1 1

T2 can be reconstructed from T0 and the number of ingoing edges (for each node)

2 3 4 5 6 7 8 9 10 11

T2

1 1

T 0 T 0

SLIDE 53

A more efficient encoding

2 3 4 5 6 7 8 9 10 11

T2

1 1 2 3 4 5 6 7 8 9 10 11

T2

1 1

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

T 0 T 0 T 0

2(n − 1) symbols= 2(n − 1) bits

00000101010100110111 T 2

(n − 1) + (n − 3) = 2n − 4 bits Canonical orderings - Schnyder woods (He, Kao, Lu ’99)

4n bits (for triangulations)

SLIDE 54

A compact data structure

u v w z u v w z 1 2 3 u v w z u v w z

u := source(e) = e/3 (w,v):=T[2e] e := (u, v)

{ (e + 1)%3

T[T[e]] (T[e] + 2)%3 u v w

Def.

z (z, v) := T[2e + 1] 0 ≤ v ≤ n − 1 0 ≤ e ≤ 3n

First simple Compact DS (size 6n)

case 1 case 2 case 3 (w, u) := color(e) = e%3 retrieve retrieving (z, u) is similar

SLIDE 55

A compact data structure

u v w z u v w z u v w z u v w z 1a 1b 2a 2b u v w z u v w z 3a 3b u v w z u v w z 3c 3d u v w z u v w z 4a 4b u v w z u v w z 4c 4d

implementation:

case analysis for black edges

More compact DS (size 5n): use minimal Schnyder woods

(less redundant and ”slightly more difficult to implement”)

many cases to distinguish

{ T[T[5u] − 3]

T[T[T[5u]]] case 1a case 1b . . . (w, u) := retrieve . . .

SLIDE 56

Half-edge Triangle-based

(non compact) data structures compact data structures

e

pposite(e)

prev(e) next(e) source(e) Winged-edge Data Structure size navigation time vertex access dynamic Half-edge/Winged-edge/Quad-edge Triangle based DS / Corner Table Directed edge (Campagna et al. ’99) 2D Catalogs (Castelli Aleardi et al., ’06) Star vertices (Kallmann et al. ’02) TRIPOD (Snoeyink, Speckmann, ’99) SOT (Gurung et al. 2010) Castelli Aleardi and Devillers (2011)

18n + n 12n + n 12n + n 7.67n 7n 6n 6n 4.8n 4n

(or 6n) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(d) O(d) O(d) O(d) O(d) yes yes yes yes no no no yes no SQUAD (Gurung et al. 2011)

(4 + ε)n

O(1) O(d) no LR (Gurung et al. 2011)

(2 + δ)n

O(1) O(d) no ε between 0.09 and 0.3 δ about 0.8 and 0.3 ESQ (Castelli Aleardi, Devillers, Rossignac’12) (or O(1))

Compact (practical) mesh data structures

Half-edge, Winged-edge, Quad-edge (19n) Triangle DS, Corner Table, Directed edge (13n) 2D Catalogs (7.67n) (7n) Star-Vertices SOT (6n) Castelli Devillers (Isaac 2011), (4n) ESQ, (4.8n)

SLIDE 57

Half-edge Triangle-based

(non compact) data structures compact data structures

e

pposite(e)

prev(e) next(e) source(e) Winged-edge Data Structure size navigation time vertex access dynamic Half-edge/Winged-edge/Quad-edge Triangle based DS / Corner Table Directed edge (Campagna et al. ’99) 2D Catalogs (Castelli Aleardi et al., ’06) Star vertices (Kallmann et al. ’02) TRIPOD (Snoeyink, Speckmann, ’99) SOT (Gurung et al. 2010) Castelli Aleardi and Devillers (2011)

18n + n 12n + n 12n + n 7.67n 7n 6n 6n 4.8n 4n

(or 6n) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(1) O(d) O(d) O(d) O(d) O(d) yes yes yes yes no no no yes no SQUAD (Gurung et al. 2011)

(4 + ε)n

O(1) O(d) no LR (Gurung et al. 2011)

(2 + δ)n

O(1) O(d) no ε between 0.09 and 0.3 δ about 0.8 and 0.3 ESQ (Castelli Aleardi, Devillers, Rossignac’12) (or O(1))

Compact (practical) mesh data structures

Half-edge, Winged-edge, Quad-edge (19n) Triangle DS, Corner Table, Directed edge (13n) 2D Catalogs (7.67n) (7n) Star-Vertices SOT (6n) Castelli Devillers (Isaac 2011), (4n) ESQ, (4.8n) vertex degree (only topological navigation)

1.2 - 1.9 times slower than Winged-edge (experimental evaluation)

(timings are expressed in nanoseconds/vertex) 4n Winged-edge 19n

50 100 150