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 4 • directed graph G = ( V , E ) 3 • level assignment ℓ : V → N s.t. ∀ ( u , v ) ∈ E : ℓ ( u ) < ℓ ( v ) 2 1

Level Graphs • 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 ) • embedding is fixed ⇐ ⇒ left-to-right order of the vertices on each level is fixed

Level Graphs • directed graph G = ( V , E ) • level assignment ℓ : V → N s.t. ∀ ( u , v ) ∈ E : ℓ ( u ) < ℓ ( v ) • embedding is fixed ⇐ ⇒ left-to-right order of the vertices on each level is fixed

λ -Drawing Model λ λ λ slope 1 λ -drawing: slope 1 / ( λ − 1) vertical

λ -Drawing Model λ λ λ slope 1 λ -drawing: slope 1 / ( λ − 1) vertical ≡ 2-drawing ≡ 3-drawing 1-drawing

λ -Drawing Model λ λ λ slope 1 λ -drawing: slope 1 / ( λ − 1) vertical ≡ 2-drawing ≡ 3-drawing 1-drawing

λ -Drawing Model

Flow Network

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

Flow Network 1 2 u e v 0 0 1 Flow on e = distance between u and v Constraint ϕ ( e ) ≥ 1 1 1 0 0 1 0 1 2 Flow = slope of dual edge Constraint 0 ≤ ϕ ( · ) ≤ 1

Flow Network 1 2 u e v 0 0 1 Flow on e = distance between u and v Constraint ϕ ( e ) ≥ 1 1 1 0 0 1 0 1 2 Lemma Flow = slope Every admissible flow corresponds of dual edge to a 2-slope drawing. Constraint 0 ≤ ϕ ( · ) ≤ 1

Flow Network 1 2 u e v 0 0 1 Flow on e = distance between u and v Constraint ϕ ( e ) ≥ 1 1 1 0 0 1 0 1 2 Lemma Flow = slope Every admissible flow corresponds of dual edge to a 2-slope drawing. max-flow: O ( n log 3 n ) Constraint 0 ≤ ϕ ( · ) ≤ 1 min-cost flow: O ( n 2 log 2 n )

Flow Network Advanced Problems: 1 2 0 0 1 • partial drawing extension (simple in connected case) 1 1 0 0 1 2

Flow Network Advanced Problems: • partial drawing extension (simple in connected case) • simultaneous drawings: given graphs G 1 , G 2 with G 1 ∩ 2 � = ∅ , are there drawings Γ 1 , Γ 2 of G 1 , G 2 s.t. G 1 ∩ 2 is drawn identically in Γ 1 , Γ 2 ? – real relaxation?

Flow Network Advanced Problems: • partial drawing v 1 / 2 1 / 2 extension (simple v ′ v 1 / 2 1 / 2 in connected case) u ′ u • simultaneous 1 / 2 1 / 2 drawings: given 1 / 2 1 / 2 graphs G 1 , G 2 with u ′ u G 1 ∩ 2 � = ∅ , are there 1 / 2 1 / 2 v ′ drawings Γ 1 , Γ 2 of G 1 , G 2 s.t. G 1 ∩ 2 is drawn identically in Γ 1 , Γ 2 ? – real relaxation?

Flow Network t 1 t 2 Advanced Problems: • partial drawing 1 1 1 v 1 / 2 1 / 2 extension (simple v ′ v 1 / 2 1 / 2 in connected case) u ′ u • simultaneous 1 / 2 1 / 2 drawings: given 1 / 2 1 / 2 graphs G 1 , G 2 with u ′ u G 1 ∩ 2 � = ∅ , are there 1 / 2 1 / 2 v ′ 1 1 drawings Γ 1 , Γ 2 of 1 G 1 , G 2 s.t. G 1 ∩ 2 is s 1 s 2 drawn identically in Γ 1 , Γ 2 ? – real relaxation?

Flow Network t 1 t 2 Advanced Problems: • partial drawing 1 1 1 v 1 / 2 1 / 2 extension (simple v ′ v 1 / 2 1 / 2 in connected case) u ′ u • simultaneous 1 / 2 1 / 2 drawings: given 1 / 2 1 / 2 graphs G 1 , G 2 with u ′ u G 1 ∩ 2 � = ∅ , are there 1 / 2 1 / 2 v ′ 1 1 drawings Γ 1 , Γ 2 of 1 G 1 , G 2 s.t. G 1 ∩ 2 is s 1 s 2 drawn identically in max. simultaneous real flow has Γ 1 , Γ 2 ? values 1 and 2, but no – real relaxation? simultaneous integer flows with these values exists

Max-Flow in Planar Graphs (w/o lower bounds) • construct directed dual G ⋆ , set ℓ ( e ⋆ ) = c ( e ) • search for shortest s ⋆ - t ⋆ path • set ϕ ( u , v ) = d ( f right ) − d ( f left ) for ( u , v ) ⋆ = ( f left , f right ) 0 | ∞ s ⋆ 12 | 12 16 | 16 0 19 | 20 1 7 | 7 12 | 16 4 9 19 s t | 0 | 5 0 0 | 4 12 4 | 4 7 | 13 11 | 14 t ⋆ 23

Max-Flow in Planar Graphs (w/o lower bounds) • construct directed dual G ⋆ , set ℓ ( e ⋆ ) = c ( e ) • search for shortest s ⋆ - t ⋆ path • set ϕ ( u , v ) = d ( f right ) − d ( f left ) for ( u , v ) ⋆ = ( f left , f right ) 0 u e v | ∞ s ⋆ 12 | 12 Flow on e = distance 16 | 16 0 between u and v 19 | 20 1 7 | 7 12 | 16 4 Constraint ϕ ( e ) ≥ 1 9 19 s t | 0 | 5 0 0 | 4 12 4 | 4 7 | 13 11 | 14 t ⋆ 23

Max-Flow in Planar Graphs (w/ lower bounds) v lower bounds on the flow: f right • definition: ϕ ( u , v ) = d ( f right ) − d ( f left ) − a • d ( f right ) ≤ d ( f left ) + b b a ≤ ϕ ( u , v ) ≤ b ⇒ ϕ ( u , v ) ≤ b • d ( f left ) ≤ d ( f right ) − a f left ⇒ ϕ ( u , v ) ≥ a u

Max-Flow and Shortest Paths

Max-Flow and Shortest Paths − 1 − 1 1 1 1 0 0 0 − 1 − 1 0 0 1 1 0 0 1 1 − 1 − 1

Max-Flow and Shortest Paths -1 0 1 − 1 − 1 • Drawing O ( n log 2 n / log log n ) 1 1 1 0 0 0 -2 -1 0 − 1 − 1 0 0 1 1 0 0 1 1 − 1 − 1 v ref -2 -1 0

Max-Flow and Shortest Paths -2 -1 1 v − 1 − 1 • Drawing O ( n log 2 n / log log n ) 1 1 1 0 0 0 -2 -1 0 − 1 − 1 − ( − 1) 0 0 1 1 − 1 0 0 1 1 − 1 − 1 v ref -2 -1 0 ( v ref , v ) : d ( v ) ≤ d ( v ref ) − 1 ⇒ d ( v ) ≤ − 1 ( v , v ref ) : d ( v ref ) ≤ d ( v ) − ( − 1) ⇒ d ( v ) ≥ − 1

Max-Flow and Shortest Paths -2 -1 1 v − 1 − 1 • Drawing O ( n log 2 n / log log n ) 0 0 1 1 1 0 -2 -1 0 − 1 − 1 − ( − 1) 0 0 1 1 − 1 0 0 1 1 − 1 − 1 v ref -2 -1 0 ( v ref , v ) : d ( v ) ≤ d ( v ref ) − 1 ⇒ d ( v ) ≤ − 1 ( v , v ref ) : d ( v ref ) ≤ d ( v ) − ( − 1) ⇒ d ( v ) ≥ − 1

Max-Flow and Shortest Paths -2 -1 1 v − 1 − 1 • Drawing O ( n log 2 n / log log n ) 0 0 1 1 • partial drawing 1 0 extension -2 -1 0 − 1 − 1 O ( n 4 / 3 log n ) 0 − ( − 1) 0 0 1 1 − 1 0 0 1 1 − 1 − 1 v ref -2 -1 0 ( v ref , v ) : d ( v ) ≤ d ( v ref ) − 1 ⇒ d ( v ) ≤ − 1 ( v , v ref ) : d ( v ref ) ≤ d ( v ) − ( − 1) ⇒ d ( v ) ≥ − 1

Max-Flow and Shortest Paths -2 -1 1 v − 1 − 1 • Drawing O ( n log 2 n / log log n ) 0 0 1 1 • partial drawing 1 0 extension -2 -1 0 − 1 − 1 O ( n 4 / 3 log n ) − ( − 1) 0 0 • simultaneous 1 1 drawings − 1 0 0 O ( n 10 / 3 log n ) 1 1 − 1 − 1 v ref -2 -1 0 the generated drawings are rightmost d 1 ( v ) < d 2 ( v ) ⇒ add constraint d 2 ( v ) ≤ d 1 ( v ) to G 2

Max-Flow and Shortest Paths -2 -1 1 v − 1 − 1 • Drawing O ( n log 2 n / log log n ) 0 0 1 1 • partial drawing 1 0 extension -2 -1 0 − 1 − 1 O ( n 4 / 3 log n ) − ( − 1) 0 0 • simultaneous 1 1 drawings − 1 0 0 O ( n 10 / 3 log n ) 1 1 − 1 − 1 • works for λ ∈ N v ref -2 -1 0 • NP-complete for “short long” edges, i.e., ℓ ( v ) − ℓ ( u ) ≤ 2

Rectilinear Planar Monotone 3- Sat x 1 ∨ x 5 ∨ x 7 x 1 ∨ x 4 ∨ x 5 x 5 ∨ x 6 ∨ x 7 x 1 ∨ x 2 ∨ x 3 x 1 x 2 x 3 x 4 x 5 x 6 x 7 ¬ x 3 ∨ ¬ x 4 ∨ ¬ x 5 ¬ x 2 ∨ ¬ x 3 ∨ ¬ x 5 ¬ x 1 ∨ ¬ x 2 ∨ ¬ x 7

Variable Gadget true true true true

Variable Gadget false false false false

(Positive) Clause Gadget true false false

(Positive) Clause Gadget false true false

(Positive) Clause Gadget false false true

(Positive) Clause Gadget false false false