Michael A. Bekos, Henry Forster, Michael Kaufmann

On Smooth Orthogonal and Octilinear Drawings:Relations, Complexity and Kandinsky Drawings

Wilhelm-Schickard-Institut fur InformatikUniversitat Tubingen, Germany

Smooth Orthogonal and Octilinear Drawings

OrthogonalComplexity 3

Extensions of the orthogonal graph drawing model

Smooth Orthogonal and Octilinear Drawings

OrthogonalComplexity 3

Smooth OrthogonalComplexity 2

Extensions of the orthogonal graph drawing model

Smooth orthogonal: Clarity of orthogonal layouts+ Aesthetics of Lombardi drawings

Smooth Orthogonal and Octilinear Drawings

OrthogonalComplexity 3

Smooth OrthogonalComplexity 2

Extensions of the orthogonal graph drawing model

Smooth orthogonal: Clarity of orthogonal layouts+ Aesthetics of Lombardi drawings

Smooth Orthogonal and Octilinear Drawings

OrthogonalComplexity 3

Smooth OrthogonalComplexity 2

Extensions of the orthogonal graph drawing model

Smooth orthogonal: Clarity of orthogonal layouts+ Aesthetics of Lombardi drawings

Smooth Orthogonal and Octilinear Drawings

OrthogonalComplexity 3

Smooth OrthogonalComplexity 2

OctilinearComplexity 2

Extensions of the orthogonal graph drawing model

Smooth orthogonal: Clarity of orthogonal layouts+ Aesthetics of Lombardi drawings

Octilinear: Generalization to max-degree 8+ Metromap applications

Known Results

Relations

Complexity

Kandinsky Drawings

Known Results

RelationsNot all max-degree 4 graphs admit bendless smoothorthogonal/octilinear drawings

Complexity

Kandinsky Drawings

1 bend per edge suffices for max-degree 4 graphs in both models

[Bekos et al. 2013, Bekos et al. 2017]

[Alam et al. 2014, Bekos et al. 2015]

Known Results

RelationsNot all max-degree 4 graphs admit bendless smoothorthogonal/octilinear drawings

ComplexityBendless octilinear drawing problem NP-hard on max-degree 8graphs

Kandinsky Drawings

1 bend per edge suffices for max-degree 4 graphs in both models

[Bekos et al. 2013, Bekos et al. 2017]

[Alam et al. 2014, Bekos et al. 2015]

[Nollenburg 2005]

Known Results

RelationsNot all max-degree 4 graphs admit bendless smoothorthogonal/octilinear drawings

ComplexityBendless octilinear drawing problem NP-hard on max-degree 8graphs

Kandinsky DrawingsBook embedding inspired approach for smooth orthogonal model(< n edges with edges of complexity 2)

1 bend per edge suffices for max-degree 4 graphs in both models

[Bekos et al. 2013, Bekos et al. 2017]

[Alam et al. 2014, Bekos et al. 2015]

[Nollenburg 2005]

[Bekos et al. 2013, Cardinal et al. 2015]

Our Contribution

Relations

Complexity

Kandinsky Drawings

Our Contribution

Relations

Classes of bendless smooth orthogonal drawable (SC1) andoctilinear drawable (8C1) graphs are incomparable

Complexity

Kandinsky Drawings

Our Contribution

Relations

Classes of bendless smooth orthogonal drawable (SC1) andoctilinear drawable (8C1) graphs are incomparable

Complexity

Deciding if a smooth orthogonal or octilinear representation isrealizable is NP-hard on max-degree 4 graphs

Kandinsky Drawings

Our Contribution

Relations

Classes of bendless smooth orthogonal drawable (SC1) andoctilinear drawable (8C1) graphs are incomparable

Complexity

Deciding if a smooth orthogonal or octilinear representation isrealizable is NP-hard on max-degree 4 graphs

Kandinsky Drawings

Smooth orthogonal: Alternative approach producing aestheticallymore pleasing drawings

Octilinear: First results

Relations

Bendless smooth orthogonal and octilineardrawings require same endpoint positioning

Relations

Bendless smooth orthogonal and octilineardrawings require same endpoint positioning

Idea: Replace arcs with diagonals and vice versa

Relations

Bendless smooth orthogonal and octilineardrawings require same endpoint positioning

Idea: Replace arcs with diagonals and vice versa

Relations

Bendless smooth orthogonal and octilineardrawings require same endpoint positioning

Idea: Replace arcs with diagonals and vice versa

But: We must retain planarity and port constraints!

Relations

Bendless smooth orthogonal and octilineardrawings require same endpoint positioning

Idea: Replace arcs with diagonals and vice versa

But: We must retain planarity and port constraints!

Relations

Bendless smooth orthogonal and octilineardrawings require same endpoint positioning

Idea: Replace arcs with diagonals and vice versa

But: We must retain planarity and port constraints!

Relations

8C3 = max-degree 8 planar8C2

SC2 = max-degree 4 planar8C1

SC1

8Ck = Graphs drawable with octilinear complexity k

SCk = Graphs drawable with smooth orthogonal complexity k

X

X

XX

??

?

Relations

8C3 = max-degree 8 planar8C2

SC2 = max-degree 4 planar8C1

SC1

8Ck = Graphs drawable with octilinear complexity k

SCk = Graphs drawable with smooth orthogonal complexity k

X

X

XX

?

??

Intersection of SC1 and 8C1

Infinitely large graph family drawable with both styles:

Intersection of SC1 and 8C1

Infinitely large graph family drawable with both styles:

Family is 4-regular → density does not divide classes

Relations

8C3 = max-degree 8 planar8C2

SC2 = max-degree 4 planar8C1

SC1

8Ck = Graphs drawable with octilinear complexity k

SCk = Graphs drawable with smooth orthogonal complexity k

X

X

XX

?

??

Relations

8C3 = max-degree 8 planar8C2

SC2 = max-degree 4 planar8C1

SC1

8Ck = Graphs drawable with octilinear complexity k

SCk = Graphs drawable with smooth orthogonal complexity k

X

X

XX

?X

?

Relations

8C3 = max-degree 8 planar8C2

SC2 = max-degree 4 planar8C1

SC1

8Ck = Graphs drawable with octilinear complexity k

SCk = Graphs drawable with smooth orthogonal complexity k

X

X

XX

?

?X

SC1 but not 8C1

Infinitely large 4-regular graph family:

SC1 but not 8C1

Infinitely large 4-regular graph family:

Two end componentsdo not admit bendlessoctilinear drawings

SC1 but not 8C1

End components only have one embedding

SC1 but not 8C1

End components only have one embedding

Properties of this embedding:

SC1 but not 8C1

End components only have one embedding

Properties of this embedding:

Each face has length at most 5

SC1 but not 8C1

End components only have one embedding

Properties of this embedding:

Each face has length at most 5

All but one vertex on the outerface must support two ports to theinterior of the drawing

SC1 but not 8C1

If we try to realize such a drawing, we find, that it is notpossible to close the outerface

SC1 but not 8C1

If we try to realize such a drawing, we find, that it is notpossible to close the outerface

SC1 but not 8C1

If we try to realize such a drawing, we find, that it is notpossible to close the outerface

SC1 but not 8C1

If we try to realize such a drawing, we find, that it is notpossible to close the outerface

SC1 but not 8C1

If we try to realize such a drawing, we find, that it is notpossible to close the outerface

Relations

8C3 = max-degree 8 planar8C2

SC2 = max-degree 4 planar

SC1

8C1

8Ck = Graphs drawable with octilinear complexity k

SCk = Graphs drawable with smooth orthogonal complexity k

X

X

XX

?

?X

Relations

8C3 = max-degree 8 planar8C2

SC2 = max-degree 4 planar

SC1

8C1

8Ck = Graphs drawable with octilinear complexity k

SCk = Graphs drawable with smooth orthogonal complexity k

X

X

XX

?X

X

Relations

8C3 = max-degree 8 planar8C2

SC2 = max-degree 4 planar

SC1

8C1

8Ck = Graphs drawable with octilinear complexity k

SCk = Graphs drawable with smooth orthogonal complexity k

X

X

XX

X?

X

8C1 but not SC1

Infinitely large 4-regular graph family:

8C1 but not SC1

Infinitely large 4-regular graph family:

Multiple copies of a basic component in a cycle

8C1 but not SC1

Infinitely large 4-regular graph family:

Multiple copies of a basic component in a cycle2 separation pairs that allow flips

8C1 but not SC1

Infinitely large 4-regular graph family:

Multiple copies of a basic component in a cycle

All but one copy must have the outerface as shown in the figure

2 separation pairs that allow flips

8C1 but not SC1

Infinitely large 4-regular graph family:

Multiple copies of a basic component in a cycle

All but one copy must have the outerface as shown in the figure

In order to connect to other copies: 2 free ports at red vertices

2 separation pairs that allow flips

8C1 but not SC1

Infinitely large 4-regular graph family:

Multiple copies of a basic component in a cycle

All but one copy must have the outerface as shown in the figure

In order to connect to other copies: 2 free ports at red vertices

2 separation pairs that allow flips

Possible embeddings are isomorphic to each other

8C1 but not SC1

Infinitely large 4-regular graph family:

Multiple copies of a basic component in a cycle

All but one copy must have the outerface as shown in the figure

In order to connect to other copies: 2 free ports at red vertices

2 separation pairs that allow flips

Case analysis: No smooth orthogonal drawing existsPossible embeddings are isomorphic to each other

Complexity

Input: angles between edges and edge segments along edges

Smooth Orthogonal Representation Realizability

Complexity

Input: angles between edges and edge segments along edges

Output: drawing realizing input constraints

Smooth Orthogonal Representation Realizability

Complexity

Input: angles between edges and edge segments along edges

Output: drawing realizing input constraints

Smooth Orthogonal Representation Realizability

Equivalent to last step of TSM [Tamassia 1987]

Complexity

Reduction from 3-SAT

Input: angles between edges and edge segments along edges

Output: drawing realizing input constraints

Smooth Orthogonal Representation Realizability

Equivalent to last step of TSM [Tamassia 1987]

Complexity

General Ideas:

Encode information in edge lengths

Reduction from 3-SAT

Input: angles between edges and edge segments along edges

Output: drawing realizing input constraints

Smooth Orthogonal Representation Realizability

Equivalent to last step of TSM [Tamassia 1987]

Complexity

General Ideas:

Encode information in edge lengths

Propagate along rectangular faces

Reduction from 3-SAT

Input: angles between edges and edge segments along edges

Output: drawing realizing input constraints

Smooth Orthogonal Representation Realizability

Equivalent to last step of TSM [Tamassia 1987]

Complexity

General Ideas:

Encode information in edge lengths

Propagate along rectangular faces

Change direction with triangular faces

Reduction from 3-SAT

Input: angles between edges and edge segments along edges

Output: drawing realizing input constraints

Smooth Orthogonal Representation Realizability

Equivalent to last step of TSM [Tamassia 1987]

Complexity

General Ideas:

Encode information in edge lengths

Propagate along rectangular faces

Change direction with triangular faces

Reduction from 3-SAT

Ensure that two sums of informationare the same

Input: angles between edges and edge segments along edges

Output: drawing realizing input constraints

Smooth Orthogonal Representation Realizability

Equivalent to last step of TSM [Tamassia 1987]

Auxiliary Gadgets

Copy gadget

Auxiliary Gadgets

Copy gadget

x

xx

x

Auxiliary Gadgets

Copy gadget

x

xx

x

Auxiliary Gadgets

Copy gadget

x

xx

x

Crossing gadget

Auxiliary Gadgets

Copy gadget

b

a

b

a

x

xx

x

Crossing gadget

Auxiliary Gadgets

Copy gadget

b

a

b

a

x

xx

x

Crossing gadget

We can connect literals and clauses properly

Variable Gadget

Variable Gadget

Take 3 units of “flow” as input

u u u

Variable Gadget

Take 3 units of “flow” as input

Any algorithm must decide how to distribute on x and x

ux

x

u u

Variable Gadget

Take 3 units of “flow” as input

Any algorithm must decide how to distribute on x and x

u

x

x

u u

Variable Gadget

Take 3 units of “flow” as input

Any algorithm must decide how to distribute on x and x

We immediately get a negation

u

x

x

u u

Variable Gadget

Take 3 units of “flow” as input

Any algorithm must decide how to distribute on x and x

We immediately get a negation

We enforce a parity between x and x with another gadget

u

x

x

u u

Variable Gadget

Take 3 units of “flow” as input

Any algorithm must decide how to distribute on x and x

We immediately get a negation

We enforce a parity between x and x with another gadget

`(true) ' 2`(u) and `(false) / `(u)

u

x

x

u u

Clause Gadget

Clause Gadget

One side of the arc is 4 units + a free edge’s length long

u u u u

Clause Gadget

a

b

c

The other side length is defined by the literals of the clause

One side of the arc is 4 units + a free edge’s length long

u u u u

Clause Gadget

a

b

c

The other side length is defined by the literals of the clause

One side of the arc is 4 units + a free edge’s length long

`(true) ' 2`(u) and `(false) / `(u)⇒ at least one literal must be true

u u u u

A “Small” Example

(a∨ b ∨ c)∧ (a∨ b ∨ c) with a = false and b = c = true

A “Small” Example

(a∨ b ∨ c)∧ (a∨ b ∨ c) with a = false and b = c = true

Copies of unit length

A “Small” Example

(a∨ b ∨ c)∧ (a∨ b ∨ c) with a = false and b = c = true

Variable a

Variable b

Variable c

Copies of unit length

A “Small” Example

(a∨ b ∨ c)∧ (a∨ b ∨ c) with a = false and b = c = true

Variable a

Variable b

Variable c

Copies of unit length

EnsureParity

EnsureParity

A “Small” Example

(a∨ b ∨ c)∧ (a∨ b ∨ c) with a = false and b = c = true

Variable a

Variable b

Variable c

Copies of unit length

EnsureParity

EnsureParity

a ∨ b ∨ c a ∨ b ∨ c

A “Small” Example

(a∨ b ∨ c)∧ (a∨ b ∨ c) with a = false and b = c = true

Variable a

Variable b

Variable c

Copies of unit length

EnsureParity

EnsureParity

a ∨ b ∨ c a ∨ b ∨ c

A “Small” Example

(a∨ b ∨ c)∧ (a∨ b ∨ c) with a = false and b = c = true

Variable a

Variable b

Variable c

Copies of unit length

EnsureParity

EnsureParity

a ∨ b ∨ c a ∨ b ∨ c

Remarks

Octilinear Representation Realizability is NP-hard onmax-degree 4 graphs

Remarks

Octilinear Representation Realizability is NP-hard onmax-degree 4 graphs

Same reduction scheme, most gagdets easy to transform

Remarks

Octilinear Representation Realizability is NP-hard onmax-degree 4 graphs

Same reduction scheme, most gagdets easy to transform

TSM approach not suitable for smooth orthogonal andoctilinear drawings

Kandinsky Drawings

Kandinsky model in smooth orthogonalsetting so far: Book Embedding Inspired

[Bekos et al. 2013]

Kandinsky Drawings

Kandinsky model in smooth orthogonalsetting so far: Book Embedding Inspired

[Bekos et al. 2013]

O(n) time, O(n2) area, ≤ n − 2 edges of complexity 2...But is it readable?

Kandinsky Drawings

Kandinsky model in smooth orthogonalsetting so far: Book Embedding Inspired

[Bekos et al. 2013]

O(n) time, O(n2) area, ≤ n − 2 edges of complexity 2...But is it readable?

Possible improvements:

Distribute vertices more evenly

Draw edges x , y -monotone

Our Modified Shift-Method

Based on [de Fraysseix, Pach, Pollack 1990]

Our Modified Shift-Method

Based on [de Fraysseix, Pach, Pollack 1990]

Contour condition: Only quarter circular arcs on Γk−1

w1 = v1

w`wr

wp = v2

Γk−1

Our Modified Shift-Method

Based on [de Fraysseix, Pach, Pollack 1990]

Contour condition: Only quarter circular arcs on Γk−1

w`wr

Γk−1

w1 = v1 wp = v2Shift blue and green vertices as in the originalshift-method, draw (w`, vk) and (wr , vk) as a single arc

vk

Our Modified Shift-Method

Based on [de Fraysseix, Pach, Pollack 1990]

We can use this approach for octilinear Kandinsky drawings too!

Contour condition: Only quarter circular arcs on Γk−1

Shift blue and green vertices as in the originalshift-method, draw (w`, vk) and (wr , vk) as a single arc

vk

w`wr

Γk−1

w1 = v1 wp = v2

vk

Drawings with Less Bends

So far: Linear time, quadratic area, bi-monotonicity, but noguarantee for number of bends

Drawings with Less Bends

So far: Linear time, quadratic area, bi-monotonicity, but noguarantee for number of bends

Improve in two steps:

Drawings with Less Bends

So far: Linear time, quadratic area, bi-monotonicity, but noguarantee for number of bends

Improve in two steps:

Cut and stretch at edges whose two endpoints have a larger y - thanx-distance

Drawings with Less Bends

So far: Linear time, quadratic area, bi-monotonicity, but noguarantee for number of bends

Improve in two steps:

Cut and stretch at edges whose two endpoints have a larger y - thanx-distance

Compute new y -coordinates

Drawings with Less Bends

So far: Linear time, quadratic area, bi-monotonicity, but noguarantee for number of bends

Improve in two steps:

Cut and stretch at edges whose two endpoints have a larger y - thanx-distance

Compute new y -coordinates

n − 1 edges without bends, O(n4) area

Drawings with Less Bends

So far: Linear time, quadratic area, bi-monotonicity, but noguarantee for number of bends

Improve in two steps:

Cut and stretch at edges whose two endpoints have a larger y - thanx-distance

Compute new y -coordinates

one

Thanks to theanonymous reviewers!n − 1 edges without bends, O(n4) area

O(n3)

Drawings with Less Bends

w1 = v1 wp = v2

Γk−1

w`

wr

Now: steep mountains as contour

Drawings with Less Bends

w1 = v1 wp = v2

Γk−1

w`

wr

Now: steep mountains as contour

Lk

x(vk) is fixed

Drawings with Less Bends

w1 = v1 wp = v2

Γk−1

w`

wr

Now: steep mountains as contour

Lk

x(vk) is fixed =⇒ ensure one edge of complexity 1

Drawings with Less Bends

w1 = v1 wp = v2

Γk−1

w`

wr

Now: steep mountains as contour

Lk

x(vk) is fixed =⇒ ensure one edge of complexity 1

Drawings with Less Bends

w1 = v1 wp = v2

Γk−1

w`

wr

Now: steep mountains as contour

Lk

x(vk) is fixed =⇒ ensure one edge of complexity 1Use highest candidate position to ensure planarity andcontour condition

Open Problems

Relations

Complexity

Kandinsky Drawings

Open Problems

Relations

Higher degree: (smooth) d-linear drawings

Complexity

Kandinsky Drawings

Relations of smooth d-linear and 2d-linear on max-degree d

Open Problems

Relations

Higher degree: (smooth) d-linear drawings

Complexity

Complexity of general bendless smooth orthogonal and octilineardrawing problems on max-degree 4 graphs

Kandinsky Drawings

Relations of smooth d-linear and 2d-linear on max-degree d

Open Problems

Relations

Higher degree: (smooth) d-linear drawings

Complexity

Complexity of general bendless smooth orthogonal and octilineardrawing problems on max-degree 4 graphs

Kandinsky Drawings

Generalize bend reduction to non-triangulated planar graphs

Bend reduction for the octilinear model

Relations of smooth d-linear and 2d-linear on max-degree d

Open Problems

Relations

Higher degree: (smooth) d-linear drawings

Complexity

Complexity of general bendless smooth orthogonal and octilineardrawing problems on max-degree 4 graphs

Kandinsky Drawings

Generalize bend reduction to non-triangulated planar graphs

Bend reduction for the octilinear model

Thanks for yourattention!

Relations of smooth d-linear and 2d-linear on max-degree d