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


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

Minimum-Area Drawings

A Straight-Line Drawing of G


on 8×5 grid

A Plane Graph G

W = 8

H = 5

W = 6

H = 4


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]


(n−2)×(n−2)

Brandenburg[2004]


(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


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


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


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


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


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


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.



c b

a

No line is available to place the representative vertex

k

le

a b

c


13

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



c b

a


a b

c


14



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



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


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


...

...

... ... ... 17

Lower Bound on Area

(2n/3 -1)

(2n/3 -1)


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.

18

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


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.

20


Documents

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