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Md. Emran Chowdhury Muhammad Jawaherul Alam Md. Saidur Rahman Department of Computer Science & Engineering Bangladesh University of Engineering & Technology (BUET)

On Upward Point-Set Embedding of Upward Planar Digraphs

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On Upward Point-Set Embedding of Upward Planar Digraphs. Md. Emran Chowdhury. Muhammad Jawaherul Alam. Md. Saidur Rahman. Department of Computer Science & Engineering Bangladesh University of Engineering & Technology (BUET). Upward Point-Set Embedding. d. d. f. f. e. e. a. a. G. - PowerPoint PPT Presentation

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Page 1: On Upward Point-Set Embedding of Upward Planar Digraphs

Md. Emran ChowdhuryMd. Emran Chowdhury

Muhammad Jawaherul AlamMuhammad Jawaherul Alam Md. Saidur RahmanMd. Saidur Rahman

Department of Computer Science & EngineeringBangladesh University of Engineering & Technology (BUET)

Department of Computer Science & EngineeringBangladesh University of Engineering & Technology (BUET)

Page 2: On Upward Point-Set Embedding of Upward Planar Digraphs

Upward Point-Set EmbeddingUpward Point-Set Embedding

a

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df

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Sa

c

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e

Each vertex is placed at a distinct pointEach vertex is placed at a distinct point

G

Page 3: On Upward Point-Set Embedding of Upward Planar Digraphs

Upward Point-Set EmbeddingUpward Point-Set Embedding

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c

b

d

a

e

a

c

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df

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Each vertex is placed at a distinct pointEach vertex is placed at a distinct point

Each edge is drawn upwardEach edge is drawn upward

G

Page 4: On Upward Point-Set Embedding of Upward Planar Digraphs

Upward Point-Set EmbeddingUpward Point-Set Embedding

S

f

c

b

d

a

e

a

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df

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G

ac

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df

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G

Each vertex is placed at a distinct pointEach vertex is placed at a distinct point

Each edge is drawn upwardEach edge is drawn upward

ac

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df

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G’ S

f

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b

d

a

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ac

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G S

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Page 5: On Upward Point-Set Embedding of Upward Planar Digraphs

a

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df

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G

Upward Point-Set EmbeddingUpward Point-Set Embedding

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S

f

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S

There is an Upward Point-set Embedding of G onS if and only if Gis acyclic

There is an Upward Point-set Embeddingof G on S if and only if G is acyclic There is an Upward Point-set Embedding

ofG on S if and only if Gis upward planar

There is an Upward Point-set Embeddingof G on S if and only if G is upward planar

Page 6: On Upward Point-Set Embedding of Upward Planar Digraphs

Upward Point-Set EmbeddingUpward Point-Set Embedding

S

S

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Page 7: On Upward Point-Set Embedding of Upward Planar Digraphs

Upward Point-Set EmbeddingUpward Point-Set Embedding

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Giordano et. al.

Upward Point-Set Embedding ofany upward planar digraph with on

any point set withat most two bends per edge

Page 8: On Upward Point-Set Embedding of Upward Planar Digraphs

a

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Upward Point-Set Embedding with mappingUpward Point-Set Embedding with mapping

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bS

G

φ

Page 9: On Upward Point-Set Embedding of Upward Planar Digraphs

ac

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G

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φ’

No upward point-setembedding with this mapping

No upward point-setembedding with this mapping

Upward Point-Set Embedding with mappingUpward Point-Set Embedding with mapping

Page 10: On Upward Point-Set Embedding of Upward Planar Digraphs

Giordano et. al.

• O(n3)-time testing algorithm

• O(n2)-time drawing algorithm* (2n-3) bends per edge

Ours

• O(n2)-time drawing algorithm* (n-3) bends per edge

Upward TopologicalBook Embedding with

a given ordering

Upward Point-Set Embedding with mappingUpward Point-Set Embedding with mapping

Page 11: On Upward Point-Set Embedding of Upward Planar Digraphs

Upward Topological Book EmbeddingUpward Topological Book Embedding

Variant of Upward Point-Set Embedding

ac

d

b

SG

Spine

LeftPage

RightPage

a

c

b

d

The vertices on the spine

The edges on the pages

Only ordering of the verticesare important, not their positions

a

c

b

d

Page 12: On Upward Point-Set Embedding of Upward Planar Digraphs

Our AlgorithmOur Algorithm

G contains directed hamiltonian pathG contains directed hamiltonian path

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A directed path containingall the vertices

A directed path containingall the vertices

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Page 13: On Upward Point-Set Embedding of Upward Planar Digraphs

Our AlgorithmOur Algorithm

G contains directed hamiltonian pathG contains directed hamiltonian path

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1

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7The drawing …..

• has no crossings sinceit has the same embeddingas the original graph

• has no spine crossings• has 1 bend per edge

Page 14: On Upward Point-Set Embedding of Upward Planar Digraphs

Our AlgorithmOur Algorithm

G contains no directed hamiltonian path

1

3

2

4

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5

a

b

cd

e

Page 15: On Upward Point-Set Embedding of Upward Planar Digraphs

Our AlgorithmOur Algorithm

G contains no directed hamiltonian path

a

b

cd

e

1

3

2

4

7

6

5

Page 16: On Upward Point-Set Embedding of Upward Planar Digraphs

Our AlgorithmOur Algorithm

a

b

cd

e

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a

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Page 17: On Upward Point-Set Embedding of Upward Planar Digraphs

a

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cd

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Our AlgorithmOur Algorithm

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Page 18: On Upward Point-Set Embedding of Upward Planar Digraphs

1

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a

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cd

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Our AlgorithmOur Algorithm

d

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Each spine crossingcorresponds to a dummy vertex

Each spine crossingcorresponds to a dummy vertex

Page 19: On Upward Point-Set Embedding of Upward Planar Digraphs

Number of Bends per edgeNumber of Bends per edge

i

i+1

i+2

j-2

j-1

j

Spine crossing from ito j is at most j-i-3

Spine crossing from ito j is at most j-i-3

The edge (1, n) has no crossingsThe edge (1, n) has no crossings

Spine Crossings per edgeis at most n-4

Spine Crossings per edgeis at most n-4

Bends per edge is at most n-3Bends per edge is at most n-3

Page 20: On Upward Point-Set Embedding of Upward Planar Digraphs

Algorithm for Points in General PositionAlgorithm for Points in General Position

e

c

a

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e

c

a

b

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a’

d’c’

b’e’

L

Ordering inducedby φ on L

φ

S

G

Giordano et. al. showed that

G admits an upward point-set embedding onS of points with the mapping φ with t bends

if and only if there is a line L such thatG admits an upward topological book embeddingwith the ordering induced by φ on L with t bends

Page 21: On Upward Point-Set Embedding of Upward Planar Digraphs

Open ProblemsOpen Problems

Find the minimum number of total bends in all edges

To give an o(n3)-time testing algorithm

Page 22: On Upward Point-Set Embedding of Upward Planar Digraphs

Thank You