[PPT] - Weak Unit Disk Contact Representations for Graphs without Embedding PowerPoint Presentation

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Weak Unit Disk Contact Representations for Graphs without Embedding

Jonas Cleve, Freie Universität Berlin

EuroCG 2020 – March 16–18 – Würzburg, Germany

SLIDE 2

Unit Disk Contact Representations

Unit disk contact representation (UDCR) of G = (V, E):

graph G 1 2 3 4 5 6 7 8

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Unit Disk Contact Representations

Unit disk contact representation (UDCR) of G = (V, E):

graph G

one unit-disk D(v) per node v s. t.

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

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Unit Disk Contact Representations

Unit disk contact representation (UDCR) of G = (V, E):

graph G

one unit-disk D(v) per node v s. t.
{u, v} ∈ E ⇐

⇒ D(u), D(v) touch.

UDCR of G 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

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Unit Disk Contact Representations

Unit disk contact representation (UDCR) of G = (V, E):

graph G

one unit-disk D(v) per node v s. t.
{u, v} ∈ E ⇐

⇒ D(u), D(v) touch.

UDCRs of G

No embedding: neighbor order can be chosen arbitrarily

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

1

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Unit Disk Contact Representations

Unit disk contact representation (UDCR) of G = (V, E):

graph G

one unit-disk D(v) per node v s. t.
{u, v} ∈ E ⇐

⇒ D(u), D(v) touch.

UDCRs of G

No embedding: neighbor order can be chosen arbitrarily

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 7 8 5 6

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Unit Disk Contact Representations

graph G

one unit-disk D(v) per node v s. t.

No embedding: neighbor order can be chosen arbitrarily

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 7 8 5 6

Weak unit disk contact representation (WUDCR) of G = (V, E):

{u, v} ∈ E =

⇒ D(u), D(v) touch.

WUDCRs of G

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Results

Existing Work: NP-hardness and construction algorithms. star caterpillar tree Weak UDCR Weak Emb. UDCR

Chiu, Cleve, Nöllenburg; 2019 Cleve; 2020 Cleve; 2020 2

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Results

Existing Work: NP-hardness and construction algorithms. star caterpillar tree Weak UDCR Weak Emb. UDCR

Chiu, Cleve, Nöllenburg; 2019 Cleve; 2020 Cleve; 2020 2

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Recognizing Caterpillars in Linear Time

A caterpillar is a tree
backbone path plus leaves

Reminder:

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Recognizing Caterpillars in Linear Time

Algorithm: Caterpillar G

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Recognizing Caterpillars in Linear Time

Algorithm: Caterpillar G Find longest path v0, v1, . . . , vk, vk+1 backbone

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Recognizing Caterpillars in Linear Time

Algorithm: Caterpillar G Find longest path v0, v1, . . . , vk, vk+1 backbone Place backbone disks horizontally

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Recognizing Caterpillars in Linear Time

Algorithm: Caterpillar G Find longest path v0, v1, . . . , vk, vk+1 backbone Place backbone disks horizontally Place leaf disks at leftmost position

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Recognizing Caterpillars in Linear Time

Algorithm: Caterpillar G Find longest path v0, v1, . . . , vk, vk+1 backbone Place backbone disks horizontally Place leaf disks at leftmost position

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Recognizing Caterpillars in Linear Time

Algorithm: Caterpillar G Find longest path v0, v1, . . . , vk, vk+1 backbone Place backbone disks horizontally Place leaf disks at leftmost position

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Recognizing Caterpillars in Linear Time

Algorithm: Caterpillar G Find longest path v0, v1, . . . , vk, vk+1 backbone Place backbone disks horizontally Place leaf disks at leftmost position

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Recognizing Caterpillars in Linear Time

Algorithm: Caterpillar G Find longest path v0, v1, . . . , vk, vk+1 backbone Place backbone disks horizontally Place leaf disks at leftmost position

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SLIDE 19

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard.

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NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

4

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NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

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NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

4

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NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine. Each clause: {0, 0, 1} or {0, 1, 1}

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NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine. Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

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SLIDE 25

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine. Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

4

SLIDE 26

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

variable poles & clause levels

4

SLIDE 27

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

variable poles & clause levels
positive literal: fmag on bottom

4

SLIDE 28

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

variable poles & clause levels
positive literal: fmag on bottom
negative literal: fmag on top

4

SLIDE 29

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

variable poles & clause levels
positive literal: fmag on bottom
negative literal: fmag on top
no literal: both fmags

x4 x3 x2

4

SLIDE 30

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

variable poles & clause levels
positive literal: fmag on bottom
negative literal: fmag on top
no literal: both fmags

x4 x3 x2

4

SLIDE 31

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3 = 1 = 1 = 1 = 1

Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

variable poles & clause levels
positive literal: fmag on bottom
negative literal: fmag on top
no literal: both fmags

x4 x3 x2

4

SLIDE 32

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3 = 1 = 1 = 1 = 1

Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

variable poles & clause levels
positive literal: fmag on bottom
negative literal: fmag on top
no literal: both fmags

4

SLIDE 33

NP-hardness of Recognizing Trees

Theorem: Given a tree T, answering “∃ weak UDCR for T?” is NP-hard. Proof: Not-All-Equal-3SAT ≤P Tree-WUDCR, via logic engine.

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3 = 0 = 1 = 1 = 1

Logic engine Each clause: {0, 0, 1} or {0, 1, 1} Example: c1 = (x1, x2, x3) c2 = (x1, x2, x4) c3 = (x1, x3, x4)

variable poles & clause levels
positive literal: fmag on bottom
negative literal: fmag on top
no literal: both fmags

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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Modeling Line Segments

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A Branching Gadget

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A Branching Gadget

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A Branching Gadget

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A Branching Gadget

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A Branching Gadget

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A Branching Gadget

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A Branching Gadget

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

⇒

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

⇒

c3 c2 c1 c1 c2 c3 x1 x2 x3 x4

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

⇒

x1 x2 x3 x4 c3 c2 c1 c3 c2 c1

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c1 c2 c3

⇒

x1 x2 x3 x4 c3 c2 c1 c3 c2 c1

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c3 c2 c1

⇐

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c3 c2 c1

⇐

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c3 c2 c1

⇐

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c3 c2 c1

⇐

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60° Logic Engine

x1 x2 x3 x4 c3 c2 c1 c3 c2 c1

⇐

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Conclusion

star caterpillar tree Weak UDCR Weak Emb. UDCR

Cleve; 2020 Cleve; 2020 Chiu, Cleve, Nöllenburg; 2019 8

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Conclusion

star caterpillar tree Weak UDCR Weak Emb. UDCR

Cleve; 2020 Cleve; 2020

Recognizing weak UDCR for caterpillars takes linear time
Recognizing weak UDCR for trees is NP-hard

Chiu, Cleve, Nöllenburg; 2019 8

SLIDE 70

Conclusion

star caterpillar tree Weak UDCR Weak Emb. UDCR

Cleve; 2020 Cleve; 2020

Recognizing weak UDCR for caterpillars takes linear time
Recognizing weak UDCR for trees is NP-hard

NP-completeness?

Chiu, Cleve, Nöllenburg; 2019 8

SLIDE 71

Conclusion

star caterpillar Weak UDCR Weak Emb. UDCR

Cleve; 2020

Recognizing weak UDCR for caterpillars takes linear time
Recognizing weak UDCR for trees is NP-hard

NP-completeness?

Lobster graphs (k = 2)?
Trees in which all vertices have distance at most k to a central path?

Cleve; 2020

tree lobster … ?

Chiu, Cleve, Nöllenburg; 2019 8