Level-Planar Drawings with Few Slopes Guido Br uckner Nadine - - PowerPoint PPT Presentation

Level-Planar Drawings with Few Slopes

Guido Br¨ uckner Nadine Krisam Tamara Mchedlidze

Level Graphs

1 2 3 4

• directed graph G = (V , E)
• level assignment ℓ : V → N s.t.

∀(u, v) ∈ E : ℓ(u) < ℓ(v)

Level Graphs

• directed graph G = (V , E)
• level assignment ℓ : V → N s.t.

∀(u, v) ∈ E : ℓ(u) < ℓ(v)

Level Graphs

• embedding is fixed

⇐ ⇒ left-to-right order of the vertices on each level is fixed

• directed graph G = (V , E)
• level assignment ℓ : V → N s.t.

∀(u, v) ∈ E : ℓ(u) < ℓ(v)

Level Graphs

• embedding is fixed

⇐ ⇒ left-to-right order of the vertices on each level is fixed

• directed graph G = (V , E)
• level assignment ℓ : V → N s.t.

∀(u, v) ∈ E : ℓ(u) < ℓ(v)

λ-Drawing Model

vertical slope 1/(λ − 1) slope 1

λ-drawing: λ λ λ

λ-Drawing Model

1-drawing ≡ 2-drawing ≡ 3-drawing

vertical slope 1/(λ − 1) slope 1

λ-drawing: λ λ λ

λ-Drawing Model

1-drawing ≡ 2-drawing ≡ 3-drawing

vertical slope 1/(λ − 1) slope 1

λ-drawing: λ λ λ

λ-Drawing Model

Flow Network

Flow Network

1 1 2 1 2 e u v Flow on e = distance between u and v Constraint ϕ(e) ≥ 1

Flow Network

1 1 1 1 2 1 2 e u v Flow on e = distance between u and v Constraint ϕ(e) ≥ 1 1 Flow = slope

• f dual edge

Constraint 0 ≤ ϕ(·) ≤ 1

Flow Network

1 1 1 1 2 1 2 e u v Flow on e = distance between u and v Constraint ϕ(e) ≥ 1 1 Flow = slope

• f dual edge

Constraint 0 ≤ ϕ(·) ≤ 1 Lemma Every admissible flow corresponds to a 2-slope drawing.

Flow Network

1 1 1 1 2 1 2 e u v Flow on e = distance between u and v Constraint ϕ(e) ≥ 1 1 Flow = slope

• f dual edge

Constraint 0 ≤ ϕ(·) ≤ 1 Lemma Every admissible flow corresponds to a 2-slope drawing. max-flow: O(n log3 n)

min-cost flow: O(n2 log2 n)

Flow Network

Advanced Problems:

• partial drawing

extension (simple in connected case) 1 1 1 1 2 1 2

Flow Network

Advanced Problems:

• partial drawing

extension (simple in connected case)

• simultaneous

drawings: given graphs G1, G2 with G1∩2 = ∅, are there drawings Γ1, Γ2 of G1, G2 s.t. G1∩2 is drawn identically in Γ1, Γ2? – real relaxation?

Flow Network

Advanced Problems:

• partial drawing

extension (simple in connected case)

• simultaneous

drawings: given graphs G1, G2 with G1∩2 = ∅, are there drawings Γ1, Γ2 of G1, G2 s.t. G1∩2 is drawn identically in Γ1, Γ2? – real relaxation? u v u′ v ′ u v v ′ u′

1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2

Flow Network

Advanced Problems:

• partial drawing

extension (simple in connected case)

• simultaneous

drawings: given graphs G1, G2 with G1∩2 = ∅, are there drawings Γ1, Γ2 of G1, G2 s.t. G1∩2 is drawn identically in Γ1, Γ2? – real relaxation? u v u′ v ′ u v v ′ u′ s1 t1 t2 s2

1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1 1 1 1 1 1

Flow Network

Advanced Problems:

• partial drawing

extension (simple in connected case)

• simultaneous

drawings: given graphs G1, G2 with G1∩2 = ∅, are there drawings Γ1, Γ2 of G1, G2 s.t. G1∩2 is drawn identically in Γ1, Γ2? – real relaxation? u v u′ v ′ u v v ′ u′ s1 t1 t2 s2

1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1 1 1 1 1 1

• max. simultaneous real flow has

values 1 and 2, but no simultaneous integer flows with these values exists

Max-Flow in Planar Graphs (w/o lower bounds)

s t 16 | 16 4 | 4 | 9 12 | 12 11 | 14 19 | 20 0 | 5 7 | 13 7 | 7 0 | 4 4 | 1 | ∞ s⋆ t⋆ 16 12 12 19 23

• construct directed dual G ⋆, set ℓ(e⋆) = c(e)
• search for shortest s⋆-t⋆ path
• set ϕ(u, v) = d(fright) − d(fleft) for (u, v)⋆ = (fleft, fright)

Max-Flow in Planar Graphs (w/o lower bounds)

s t 16 | 16 4 | 4 | 9 12 | 12 11 | 14 19 | 20 0 | 5 7 | 13 7 | 7 0 | 4 4 | 1 | ∞ s⋆ t⋆ 16 12 12 19 23

• construct directed dual G ⋆, set ℓ(e⋆) = c(e)
• search for shortest s⋆-t⋆ path
• set ϕ(u, v) = d(fright) − d(fleft) for (u, v)⋆ = (fleft, fright)

e u v Flow on e = distance between u and v Constraint ϕ(e) ≥ 1

Max-Flow in Planar Graphs (w/ lower bounds)

a ≤ ϕ(u, v) ≤ b b −a u v fleft fright lower bounds on the flow:

• definition:

ϕ(u, v) = d(fright)−d(fleft)

• d(fright) ≤ d(fleft) + b

⇒ ϕ(u, v) ≤ b

• d(fleft) ≤ d(fright) − a

⇒ ϕ(u, v) ≥ a

Max-Flow and Shortest Paths

Max-Flow and Shortest Paths

−1 −1 −1 −1 −1 1 1 1 1 −1 1 1 1

Max-Flow and Shortest Paths

• 1

−1 −1 −1 −1 −1 1

• 1

1

• 1
• 2
• 2

1 1 1 −1 1

vref

1 1

• Drawing

O(n log2 n/ log log n)

Max-Flow and Shortest Paths

• 1

−1 −1 −1 −1 −1 1

• 2

1

• 1
• 2
• 2

1 1 1 −1 1

vref

1 1

• 1

−1 −(−1) v (vref, v) : d(v) ≤ d(vref) − 1 ⇒ d(v) ≤ −1 (v, vref) : d(vref) ≤ d(v) − (−1) ⇒ d(v) ≥ −1

• Drawing

O(n log2 n/ log log n)

Max-Flow and Shortest Paths

• 1

−1 −1 −1 −1 −1 1

1

• 1
• 2
• 2

1 1 1 1

vref (vref, v) : d(v) ≤ d(vref) − 1 ⇒ d(v) ≤ −1 (v, vref) : d(vref) ≤ d(v) − (−1) ⇒ d(v) ≥ −1

• Drawing

O(n log2 n/ log log n)

1 1 −1

• 2
• 1

v −1 −(−1)

Max-Flow and Shortest Paths

• 1

−1 −1 −1 −1 −1 1

1

• 1
• 2
• 2

1 1 1 1

vref (vref, v) : d(v) ≤ d(vref) − 1 ⇒ d(v) ≤ −1 (v, vref) : d(vref) ≤ d(v) − (−1) ⇒ d(v) ≥ −1

• Drawing

O(n log2 n/ log log n)

1 1 −1

• 2
• 1

v −1 −(−1)

• partial drawing

extension O(n4/3 log n)

Max-Flow and Shortest Paths

• 1

−1 −1 −1 −1 −1 1

1

• 1
• 2
• 2

1 1 1 1

vref

• Drawing

O(n log2 n/ log log n)

1 1 −1

• 2
• 1

v −1 −(−1)

• partial drawing

extension O(n4/3 log n) the generated drawings are rightmost d1(v) < d2(v) ⇒ add constraint d2(v) ≤ d1(v) to G2

• simultaneous

drawings O(n10/3 log n)

Max-Flow and Shortest Paths

• 1

−1 −1 −1 −1 −1 1

1

• 1
• 2
• 2

1 1 1 1

vref

• Drawing

O(n log2 n/ log log n)

1 1 −1

• 2
• 1

v −1 −(−1)

• partial drawing

extension O(n4/3 log n)

• simultaneous

drawings O(n10/3 log n)

• works for λ ∈ N
• NP-complete for

“short long” edges, i.e., ℓ(v) − ℓ(u) ≤ 2

Rectilinear Planar Monotone 3-Sat

x2 x3 x4 x5 x6 x7 x5 ∨ x6 ∨ x7 x1 ∨ x2 ∨ x3 x1 ∨ x4 ∨ x5 x1 ∨ x5 ∨ x7 ¬x3 ∨ ¬x4 ∨ ¬x5 ¬x2 ∨ ¬x3 ∨ ¬x5 ¬x1 ∨ ¬x2 ∨ ¬x7 x1

Variable Gadget

true true true true

Variable Gadget

false false false false

(Positive) Clause Gadget

true false false

(Positive) Clause Gadget

true false false

(Positive) Clause Gadget

false false true

(Positive) Clause Gadget

false false false