Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman and Md. Jawaherul Alam Graph Drawing and Information Visualization Laboratory Department 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
21
Embed
Debajyoti Mondal , Rahnuma Islam Nishat , Md. Saidur Rahman and Md. Jawaherul Alam
Minimum-Area Drawings of Plane 3-Trees. Debajyoti Mondal , Rahnuma Islam Nishat , Md. Saidur Rahman and Md. Jawaherul Alam. Graph Drawing and Information Visualization Laboratory Department of Computer Science and Engineering Bangladesh University of Engineering and Technology (BUET) - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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.
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