Beyond Level Planarity 24th International Symposium on Graph Drawing - - PowerPoint PPT Presentation

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GD ’16 – Beyond Level Planarity GD ’16 – Beyond Level Planarity

Beyond Level Planarity

24th International Symposium on Graph Drawing & Network Visualization

19–21 September, Athens, Greece

Patrizio Angelini, Giordano Da Lozzo, Fabrizio Frati, Giuseppe Di Battista, Maurizio Patrignani, Ignaz Rutter

WILHELM-SCHICKHARD-INSTITUT F ¨

UR INFORMATIK · T ¨ UBINGEN UNIVERSITY

DEPARTMENT OF ENGINEERING · ROMA TRE UNIVERSITY FACULTY OF INFORMATICS · KARLSRUHE INSTITUTE OF TECHNOLOGY

GD ’16 – Beyond Level Planarity

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GD ’16 – Beyond Level Planarity

Outline

Extensions of Level Planarity

• Level Embeddings on Surfaces
• Simultaneous Level Planarity
• Cyclic and Torus Level

Planarity

• Radial, Cyclic, and

Torus T -Level Planarity Problems: Consecutivity constraints:

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GD ’16 – Beyond Level Planarity

Level Embeddings on Surfaces (CYCLIC AND TORUS LEVEL PLANARITY)

“in order to enlarge the class of level graphs that allow for a level embedding (level drawing with no crossings), the notion of Level Planarity has been extended to surfaces different from the plane”

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GD ’16 – Beyond Level Planarity

Levels on the PLANE

ℓ3 = I × {3}

ℓ1 ℓ2 ℓ3 ℓk ℓ4

P = I × I

(V, E, γ), γ : V → {1, 2, ..., k}

edges (u, v) with γ(u) < γ(v) are not allowed

• Di Battista, G., Nardelli, E.: Hierarchies and planarity theory. IEEE Trans. Systems,

Man, and Cybernetics, 1988.

• J¨

unger,M.,Leipert,S.,Mutzel,P .: Level planarity testing in linear time. GD, 1998.

Level Planarity

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GD ’16 – Beyond Level Planarity

Levels on the SPHERE/STANDING CYLINDER

axis

S = I × S1

(V, E, γ), γ : V → {1, 2, ..., k}

edges (u, v) with γ(u) < γ(v) are not allowed (we can now draw edges that “wrap around” the cylinder axis)

ℓ3 = S1 ×

2 k−1

• Bachmaier, C., Brandenburg, F

., Forster, M.: Radial level planarity testing and embedding in linear time. JGAA, 2005.

• Fulek, R., Pelsmajer, M., Schaefer, M.: Hanani-Tutte for Radial Planarity II. GD ’16

S2

= Radial Level Planarity

ℓ1 ℓ2 ℓ3 ℓk ℓ4

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GD ’16 – Beyond Level Planarity

Levels on the ROLLING CYLINDER

ℓ3 = I × {e2πi 2

k }

ℓ2 ℓ3 ℓ1

axis

ℓk

e2πi 2 k

R = S1 × I

(V, E, γ), γ : V → {1, 2, ..., k}

edges (u, v) with γ(u) < γ(v) are allowed

Cyclic Level Planarity

• Bachmaier, C., Brunner, W.: Linear time planarity testing and embedding of strongly

connected cyclic level graphs. ESA, 2008.

• General instances? Bachmaier, Brunner, and K¨
• nig [GD’07] claimed that an O(|V|6)-time

algorithm for Cyclic LP can be obtained from the LP testing algorithm by Healy and Kuusik

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GD ’16 – Beyond Level Planarity

Levels on the

axis e2πi 2 k

ℓ1 ℓ2 ℓ3 ℓk ℓ3 = S1 × {e2πi 2

k }

T = S1 × S1

(V, E, γ), γ : V → {1, 2, ..., k}

edges (u, v) with γ(u) < γ(v) are allowed

Torus Level Planarity

• Bachmaier, C., Brunner, W., K¨
• nig, C.: Cyclic Level Planarity Testing and
• Embedding. GD ’07.
• Brunner, W.: Cyclic Level Drawings of Directed Graphs. PhD thesis, 2010.
• Hammersen, K.: A Characterization of Radial Graphs. Deutschen

Nationalbibliothek, 2013.

• pen

problem in:

?

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GD ’16 – Beyond Level Planarity

Level Planarity Variants

Lemma CYCLIC AND RADIAL LEVEL PLANARITY ≤L TORUS LEVEL PLANARITY

Level Cyclic Level Radial Level Torus Level

A (planar) level graph that is neither cyclic nor radial level planar, yet it is torus level planar

+ =

ℓ2 ℓ1

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GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: basic concepts 1/2

O′ is the

restriction of

O on φ[S] ⊆ A O′ is the

image of OS via φ

O = a1, . . . , a|A| O′ = φ(s1), . . . , ψ(s|S|) OS =

s1, . . . , s|S|

. . . . . .

• Orders and Suborders
• finite sets A (

) and S ( )

• injective map φ : S → A
• order O on A
• order OS on S
• rder O extends order OS

when: Lemma Proper Torus Level Planarity ≤L Simultaneous PQ-Ordering

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GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: basic concepts 2/2

P-nodes: permutations

• PQ-representable orders: (circular) PQ-trees represent (circular) orders
• f their leaves with consecutivity constraints
• two versions: rooted [Booth&Lueker, ’76]; unrooted: [Hsu&McConnell, ’01]
• two types of internal nodes

PQ-tree T with Leaves(T) = A

Q-nodes: flips

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GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: problem definition

arc (T1, T2, φ) is satisfied by orders O1 of T1 and O2 of T2 if

O1 extends O2

arc (T1, T2, φ)

T1 T2

φ

• input: a DAG = (N, Z)
• each node Ti ∈ N is a PQ-tree
• each arc

− →

TiTj ∈ Z is equipped with an injective map

φ : Leaves(Tj) → Leaves(Ti)

source target

• question: do there exist orders Oi for each PQ-tree Ti ∈ N that

simultaneously satisfy all the arcs in Z?

T6 T8 T5 T2 T1 T7 T3 T4

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GD ’16 – Beyond Level Planarity

From k levels to 2 levels

ℓ1 ℓ2

}

Radial Level Embedding of G1,2 = (Vi ∪ Vi+1, (Vi × Vi+1) ∩ E, γ)

ℓ1 ℓ2

Torus Level Embedding of G = (k

i=1 Vi, E, γ)

Observation:

A proper level graph G = (k

i=1 Vi, E, γ) has a torus level embedding

with orders O1, . . . , Ok on V1, . . . , Vk along ℓ1, . . . , ℓk if and only if

∃ radial level embedding of level graphs (Vi ∪ Vi+1, (Vi × Vi+1) ∩ E, γ)

• n two levels with orders O1 (Oi+1) on Vi (Vi+1) along ℓi (ℓi+1)

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GD ’16 – Beyond Level Planarity

Orders in Radial Level Embeddings:vertex ordering

ℓ1 ℓ2

Orders along ℓ1 Orders along ℓ2

Radial Level Embedding Γ

O1 on V1 O+

1 on V + 1

O2 on V2 O−

2 on V − 2

Level graph G1,2 = (V1 ∪ V2, E1,2 = E ∩ V1 × V2, γ) between ℓ1 and ℓ2

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GD ’16 – Beyond Level Planarity

Orders in Radial Level Embeddings: edge ordering

ℓ1 C ℓ2

Radial Level Embedding Γ edge ordering on E1,2 in Γ circular

• rder in which the edges inter-

sect curve C vertex-consecutive order circular

• rder O on E1,2 s.t. ∀v ∈ V1 ∪

V2 the edges incident to v are consecutive in O Level graph G1,2 = (V1 ∪ V2, E1,2 = E ∩ V1 × V2, γ) between ℓ1 and ℓ2

is

Observation: vertex-consecutive orders (and hence edge orders) are PQ-representable

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GD ’16 – Beyond Level Planarity

Radial Level Planarity on 2 Levels

ℓ1 ℓ2 ℓ1 C ℓ2

Oi := circular order on Vi O := circular order on E

Lemma

∃ RLE of G1,2 with edge ordering O in which O1 and O2 are the orders on V1 and V2

if and only if

• order O is vertex-consecutive
• orders O1 and O2 extend the orders O+

1 and O− 2 on V + 1 and V − 2

induced by O

Level graph G1,2 = (V1 ∪ V2, E1,2 = E ∩ V1 × V2, γ) between ℓ1 and ℓ2

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GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: from 2 to k levels

Instance I(G1,2) T +

1

T1,2

ι φ+

1

T −

2

φ−

2

T2

ι

T1

level tree T1:=

• univ. PQ-tree on V1

level tree T2:=

• univ. PQ-tree on V2

consistency tree T −

2 :=

• univ. PQ-tree on V −

2

consistency tree T +

1 :=

• univ. PQ-tree on V +

1

2 levels k levels Radial Torus

layer tree T1,2 represents exactly the (vertex-consecutive) edge orderings

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GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: from 2 to k levels

Instance I(G1,2) T +

1

T1,2

ι φ+

1

T −

2

φ−

2

T2

ι

T1

level tree T1:=

• univ. PQ-tree on V1

level tree T2:=

• univ. PQ-tree on V2

consistency tree T −

2 :=

• univ. PQ-tree on V −

2

consistency tree T +

1 :=

• univ. PQ-tree on V +

1

Ti,i+1

φ+

i

T −

i+1

φ−

i+1

Ti+1

ι

I(Gi,i+1)

. . . . . .

T +

i+1

ι

T −

i

Ti−1,i

ι φ−

i

Ti+1,i+2

φ+

i+1

T +

i

Ti

ι

. . . . . .

T −

k

T +

k

Tk Tk,1 Tk−1,k

ι ι φ+

k

φ−

k

T −

1

T +

1

T1 T1,2

ι ι φ+

1

Instance I(G) = k

i=1 I(Gi,i+1) (where k+1=1)

2 levels k levels

I(G) can be tested in P-time ... I(G) can be tested in P-time ...

Radial Torus

layer tree T1,2 represents exactly the (vertex-consecutive) edge orderings

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GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering:

T ′ T

φ φ(a)

a b

φ(b) φ(c)

c

µ′ µ

Fixedness

T ′

1

P-node

µ ∈ T

T ′

2

T ′

r

. . .

T1, T2, . . . , Tω, Tω+1, . . . , Tk

{ {

fixing µ not fixing µ

T

P-node µ′ in T ′ (parent) is fixed by node µ in T (child) if there exist vertex-disjoint paths:

• 1. a → µ, b → µ, c → µ in T and
• 2. φ(a) → µ′, φ(b) → µ′, φ(c) → µ′ in T ′
• some nodes (ω) in a children might fix a node in

the parent

• normalization: we can assume that all nodes in a

children fix exactly a node in the parent

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GD ’16 – Beyond Level Planarity

Simultaneous PQ-Ordering: Fixedness

Ti,i+1 T− i+1 Ti+1

. . . . . .

T− i Ti−1,i Ti+1,i+2 T+ i Ti

children fixig µ P-node µi ∈ T ′

i fixed by µ

T+ i+1

. . . . . .

2-fixed!!

• Th. 3.2,3.3 [Bl¨

asius & Rutter, SODA ’13]

• Sim. PQ-Ordering is solvable in quadratic

time for fixed(µ) ≤ 2 instances. fixedness fixed(µ) = ω + r

i=1(fixed(µi) − 1)

ω = 0 and r = 2

sink PQ-trees (consistency trees)

ω = 2 and r = 0

source PQ-trees (layer and level trees) 2-fixed!! 2-fixed!! Theorem Torus Level Planarity (Cyclic Level Planarity) can be decided in O(|V|2) time for proper level graphs (O(|V|4) time for general level graphs)

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GD ’16 – Beyond Level Planarity

Cyclic, Radial, and Torus T -Level Planarity

Cyclic T -level Planarity Radial T -Level Planarity Torus T -Level Planarity

P NPC

T -level

hint: in the construction of I(G) replace each level tree Ti with PQ-tree T i proper Instaces

Proper Cyclic T -level Planarity Proper Radial T -Level Planarity Proper Torus T -Level Planarity Proper

T -level

• input: level graph (k

i=1 Vi, E, γ) and set T = k i=1 Ti of PQ-trees with Leaves(Ti) = Vi

• question: ∃ (Cyclic) Torus Level Embedding Γ in which ∀i the order Oi on Vi along ℓi is

consistent with Ti?

• Wotzlaw, A., Speckenmeyer, E., Porschen, S.: Generalized k-ary tanglegrams
• n level graphs: A satisfiability-based approach and its evaluation. DAM,

2012.

• Angelini, P

., Da Lozzo, G., Di Battista, G., Frati, F., Roselli, V.: The importance of being proper: (in clustered-level planarity and T-level planarity). TCS, 2015.

Problem already studied in the plane: T3 T2 T1

ℓ1 ℓ2 ℓ3

computational complexity:

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GD ’16 – Beyond Level Planarity

Drawing Multiple Level Planar Graphs (SIMULTANEOUS LEVEL PLANARITY)

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GD ’16 – Beyond Level Planarity

Simultaneous Embedding with Fixed Edges =

Γ2 Γ1

+

Problem definition

• 1. vertices:
• ∀v ∈ V, Γi(v) = Γj(v)
• 2. shared edges:
• ∀e ∈ Ei ∩ Ej, Γi(e) = Γj(e)

such that

• input: G1(V, E1), G2(V, E2)
• question: existence of planar

drawings Γ1 and Γ2 SEFE

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GD ’16 – Beyond Level Planarity

Combining Problems

• 1. S = Orthogonal Drawings → Angelini, P

., Chaplick, S., Cornelsen, S., Da Lozzo, G., Di Battista, G., Eades, P ., Kindermann, P ., Kratochv` ıl, J., Lipp, F ., Rutter, I.. Simul- taneous Orthogonal Planarity. GD ’16.

• S ∈ {Upward Drawings, Convex Drawings, Level Drawings, ...}?

NEW PROBLEMS:= SEFE + Drawing Style S

Examples

OrthoSEFE of

(V, E1), (V, E2)

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GD ’16 – Beyond Level Planarity

Simultaneous Level Planarity

input: h proper level graphs (V, E1, γ), . . . , (V, Eh, γ) (γ is the same ∀ i) question: are there level embeddings Γi for each (V, Ei, γ) mapping the vertices in V to the same points along the corresponding levels? Problem definition:

• Sim. Level Embedding of (V, E1, γ), (V, E2, γ)

ℓ1 ℓ2 ℓ3 ℓ4

P 2

≥ 3

NPC

# Graphs

2

≥ 3

# Levels

NPC NPC Complexity

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GD ’16 – Beyond Level Planarity

2 Levels 2 Graphs

Cyclic Level Embedding Simultaneous Level Embedding flip Theorem

• Sim. Level Planarity of 2 graphs on 2

levels can be tested in quadratic time

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GD ’16 – Beyond Level Planarity

The Betweenness Problem

• question: existence of a linear ordering O on A such that, for each triple

ti ∈ C, it holds either

O = . . . , αi, . . . , β β βi, . . . , δi, . . . or O = . . . , δi, . . . , β β βi, . . . , αi, . . .

• input: pair A, C
• a finite set A of n objects
• a set C of m ordered triples ti = αi, β

β βi, δi of objects in A

}

t3 = 7, 2, 5 t2 = 1, 8, 6 t1 = 5, 4, 2 t4 = 7, 1, 3

A

C

} 3 5 4 6 8 1 2 7

}

Theorem [Opatrny - J. Comp.’79] Betweenness is NP-Complete

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GD ’16 – Beyond Level Planarity

Gadgets

Property:

∀i = 1, . . . , |C|, vertex uβ,j

is between uα,j and uγ,j along ℓ1

u1,1 u1,n v1,1 v1,n um,1 um,n

ℓ2 ℓ1

uα,i uγ,i

ℓ2 or ℓ0 ℓ1

Triplet gadget: single level graph (V, E, γ) on two levels Ordering gadget: pair (V, E1, γ), (V, E2, γ) of level graphs on two levels uβ,i

Property:

∀i = 1, . . . , |A|, vertices

ui,1, . . . , ui,n appear in the same left-to-right order

uj,1 uj,n x1 y1 xi yi xm ym

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GD ’16 – Beyond Level Planarity

NP-completeness

ℓ0

x1 y1 xi yi xm ym

Theorem Simultaneous Level Planarity is NP-complete even for 3 graphs on 2 levels and 2 graphs on 3 levels

ℓ1

x1 y1 xi yi xm ym

ℓ2

u1,1 u1,n um,1 um,n uj,1 uj,n u1,1 u1,n v1,1 v1,n um,1 um,n

ℓ2 ℓ1

uj,1 uj,n

• all graphs can be made connected at the expense of using one additional level
• all theorems hold true even if the simultaneous embedding is geometric, without

fixed edges, or with fixed edges

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GD ’16 – Beyond Level Planarity

Conclusions and Open Problems Results:

• we gave a simple testing and embedding algorithm for TORUS LEVEL

PLANARITY (CYCLIC LEVEL PLANARITY) that runs in O(|V|2) time for proper level graphs (O(|V|4) time for general level graphs)

• we established a complexity dichotomy for SIMULTANEOUS LEVEL

PLANARITY w.r.t. the # of graphs and the # of levels

Open problems:

• design new techniques to improve the time bounds
• extend the concept of level planarity to surfaces of higher genus

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GD ’16 – Beyond Level Planarity

ευχαριστώ

Thank you!