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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
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de
fg
hi
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k
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A plane 3-tree G
Previous ResultsPlane 3-tree
fg
hi
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k
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oa
b
c
de
CCCG 2010 August 11, 2010 6
a
b
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fg
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A plane 3-tree G
Previous ResultsProperties of Plane 3-trees
fg
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de
CCCG 2010 August 11, 2010
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The representative vertex of G is the vertex which is neighbor of all the three outer vertices of G.
The representative vertex of G
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hi
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A plane 3-tree
A plane 3-tree
A plane 3-tree
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Previous ResultsOur Idea : Dynamic Programming
CCCG 2010 August 11, 2010
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fg
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A plane 3-tree G
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Previous ResultsLet’s Try a Simpler Problem
CCCG 2010 August 11, 2010
a
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a b
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No line is available to place the vertex l
a b
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Previous ResultsLet’s Try a Simpler Problem
CCCG 2010 August 11, 2010
a b
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No line is available to place the representative vertex e
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k Let’s check whether this small plane 3-tree admits a drawing with this placement of a, b and c or not
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Is Drawr(ay, by, cy) = True ?
Previous ResultsProblem Formulation
CCCG 2010 August 11, 2010
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Representative vertex e 1 3 3
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No line is available to place the representative vertex e
Drawe(1, 2, 2) = False
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le
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Drawe(1, 3, 3) = True
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le
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Previous ResultsRecursive Solution
CCCG 2010 August 11, 2010
Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0.
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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
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No line is available to place the representative vertex
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le
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Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0.
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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
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Drawr(ay, by, cy) = False , if max{ay, by, cy} − min{ay, by, cy} = 0.
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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)
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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)
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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.
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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