Debajyoti Mondal , Rahnuma Islam Nishat , Md. Saidur Rahman and Md. Jawaherul Alam

Debajyoti Mondal, Rahnuma Islam Nishat,Md. Saidur Rahman and Md. Jawaherul Alam

Graph Drawing and Information Visualization LaboratoryDepartment of Computer Science and Engineering

Bangladesh University of Engineering and Technology (BUET)Dhaka – 1000, Bangladesh

CCCG 2010 August 11, 2010

Minimum-Area Drawings of Plane 3-Trees

Minimum-Area Drawings

A Straight-Line Drawing of G

A Straight-Line Grid-Drawing of G

on 8×5 grid

A Plane Graph G

W = 8

H = 5

W = 6

H = 4

A Straight-Line Grid-Drawing of G

on 6×4 grid CCCG 2010 August 11, 2010 2

Minimum-Area Drawings

A Straight-Line Drawing of G

A Straight-Line Grid-Drawing of G

on 8×5 grid

A Plane Graph G

W = 8

H = 5

W = 6

H = 4

A Straight-Line Grid-Drawing of G

on 6×4 grid W = 8

H = 5

W = 6

H = 4

W = 7

H = 5

W = 6

H = 6

A Minimum-Area Drawing of G

CCCG 2010 August 11, 2010 3

Previous Results de Fraysseix et al.

[1990]Straight- line grid-drawing

of plane graphs with n vertices (2n−4)×(n−2)

Schnyder[1990]

Straight- line grid-drawing of plane graphs with n vertices (n−2)×(n−2)

Brandenburg[2004]

Straight- line grid-drawing of plane graphs with n vertices

(4n/3) × (2n/3)

Krug and Wagner[2008]

Whether a planar graph has a drawing on a given area NP-Complete

This Presentation Whether a ‘plane 3-tree’ has a drawing on a given area P

CCCG 2010 August 11, 2010 4

Previous Results de Fraysseix et al.

[1990]Straight- line grid-drawing

of plane graphs with n vertices(2n−4)×(n−2)

Schnyder[1990]

Straight- line grid-drawing of plane graphs with n vertices

(n−2)×(n−2)

Brandenburg[2004]

Straight- line grid-drawing of plane graphs with n vertices

(4n/3) × (2n/3)

Krug and Wagner[2008]

Whether a plane graph has a drawing on a given area

NP-Complete

This Presentation Whether a ‘plane 3-tree’ has a drawing on a given area

P

Our Result

We obtain minimum-area drawings for plane 3-trees in polynomial time

CCCG 2010 August 11, 2010 5

a

b

c

de

fg

hi

j

k

l

mn

o

A plane 3-tree G

Previous ResultsPlane 3-tree

fg

hi

j

k

l

mn

oa

b

c

de

CCCG 2010 August 11, 2010 6

a

b

c

de

fg

hi

j

k

l

mn

o

A plane 3-tree G

Previous ResultsProperties of Plane 3-trees

fg

hi

j

k

l

mn

oa

b

c

de

CCCG 2010 August 11, 2010

c

The representative vertex of G is the vertex which is neighbor of all the three outer vertices of G.

The representative vertex of G

o

c

g

mn

d

f

hi

j

d

k

le

A plane 3-tree

A plane 3-tree

A plane 3-tree

7

Previous ResultsOur Idea : Dynamic Programming

CCCG 2010 August 11, 2010

a

b

c

de

fg

hi

j

k

l

mn

o

c

o

c

g

mn

d

f

hi

j

d

k

le

A plane 3-tree G

8

Previous ResultsLet’s Try a Simpler Problem

CCCG 2010 August 11, 2010

a

b

c

a b

c

b c

a

a c

b 9

No line is available to place the vertex l

a b

e

c

k

le

a

b

c

Previous ResultsLet’s Try a Simpler Problem

CCCG 2010 August 11, 2010

a b

c

No line is available to place the representative vertex e

a

e

c

e

k Let’s check whether this small plane 3-tree admits a drawing with this placement of a, b and c or not

k

l

a

be

c

k l

10

Is Drawr(ay, by, cy) = True ?

Previous ResultsProblem Formulation

CCCG 2010 August 11, 2010

a

be

c

k l

321

Representative vertex e 1 3 3

c b

a

No line is available to place the representative vertex e

Drawe(1, 2, 2) = False

k

le

a b

c

11

Drawe(1, 3, 3) = True

k

le

a b

c

Previous ResultsRecursive Solution

CCCG 2010 August 11, 2010

Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0.

12

Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 1 and r is an internal vertex.

Previous ResultsRecursive Solution

CCCG 2010 August 11, 2010

c b

a

No line is available to place the representative vertex

k

le

a b

c

Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0.

13

Drawr(ay, by, cy) = True , if max{ay, by, cy} − min{ay, by, cy} ≥ 1 and r is a dummy vertex.

Previous ResultsRecursive Solution

CCCG 2010 August 11, 2010

c b

a

Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 1 and r is an internal vertex.

a b

c

Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0.

14

Previous ResultsRecursive Solution

CCCG 2010 August 11, 2010

Drawr(ay, by, cy) = True , if max{ay, by, cy} − min{ay, by, cy} ≥ 1 and r is a dummy vertex.

Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 1 and r is an internal vertex.

Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0.

r

a b

c

Drawr(ay, by, cy) = vry {Drawr(ay, by, ry) & Drawr(by, cy, ry) & Drawr(cy, ay, ry)}, otherwise.

O(1)

O(1)

O(1)

h O(h)

15

Previous ResultsComplexity Analysis

CCCG 2010 August 11, 2010

Drawr(ay, by, cy)

h

O(h)

...

O(h) O(h)O(n) × × × = O(nh3)

O(nh3) × O(h) = O(nh4)

O(nh4) × O(hmin) = O(nh5min)

hmin

Computation of each entry is obtained in O(h) time.

O(nh4min)

16

false

false

false

false

A plane 3-tree G

A plane 3-tree G

A minimum-area grid

drawing of G

Patch the drawings of the subproblems to obtain the final

drawing.

false

false

Minimum-Area Grid Drawings of Plane 3-Trees

CCCG 2010 August 11, 2010

...

...

... ... ... 17

Lower Bound on Area

(2n/3 -1)

(2n/3 -1)

CCCG 2010 August 11, 2010

There exist plane graphs with n vertices that takes ⌊2(n-1)/3 ×⌋ ⌊2(n-1)/3 ⌋ area in any straight-line grid drawing.

Nested triangles graph

Frati et al.[2008]: There exist plane graphs with n vertices, n is a multiple of three, that takes (2n/3-1) ×(2n/3) area in any straight-line grid drawing.

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Minimum-area grid drawings.

Input plane 3-trees.

We observe that there exist plane 3-trees with n ≥ 6 vertices that takes

⌊2n/3 -1 × 2⌋ ⌈n/3 ⌉ area in any straight-line grid drawing.CCCG 2010 August 11, 2010

Lower Bound on Area: ⌊2(n-1)/3 ×⌋ ⌊2(n-1)/3 ⌋ ⌊2n/3 -1 × 2⌋ ⌈n/3 ⌉

When n is a multiple of three, this bound is the same as the

one by Frati et al.

19

Future Works

CCCG 2010 August 11, 2010

Devising a simpler algorithm to obtain minimum area drawings of plane 3-trees.

Determining the minimum area drawings for the other plane graphs with bounded treewidth.

Determining the area lower bound of straight-line grid drawings of planar 3-trees when the outer face is not

fixed.

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CCCG 2010 August 11, 2010