Post on 01-Jan-2016
transcript
GD 2014September 26,
2014
Drawing Planar Graphs with Reduced Height
Department of Computer ScienceUniversity of Manitoba
Stephane Durocher Debajyoti Mondal
3
Straight-line Drawings (Fixed Vs. Variable)
a
b c
d
f
GD 2014 September 26, 2014
e
g
A straight-line drawing of G on a 5×5 grid
A planar graph GA straight-line drawing
of G with height 4
a
b
c
f
g
e
eg
f
a
b
c
Straight-line Drawings (Fixed Vs. Variable)
a
b c
d
f e
g
A planar graph G
Via visibility representationBiedl, 2014
Here we allow variable embedding.Sometimes we use edge-bends.
Focus on height.
a
b
c
f
g e
Nested Triangles GraphArea Height 0.44n2 +O(1) 0.66n [Dolev et al. 1984]
A class of planar 3-trees Area Height 0.44n2 + O(1) 0.66n [Frati and
Patrignani 2008, Mondal et al. 2010]
Triangulations
Area Height2n2 + O(1) n − 2 [de Fraysseix et al. 1990]n2 + O(1) n − 2 [Schnyder 1990]1.78n2 + O(1) 0.66n [Chrobak and Nakano 1998]0.88n2 + O(1) 0.66n [Brandenburg 2008]0.44n2 + O(1) 0.66n (polyline) [Bonichon et al. 2003]
Upper Bounds Lower BoundsFixed Embedding
TriangulationsPolyline drawing with height 4n/9+O(λ∆) ≈ 0.44n+O(λ∆)This is 0.44n+o(n) when ∆ is o(n)
Planar 3-treesStraight-line drawing with height 4n/9+O(1) ≈ 0.44n+O(1)
TriangulationsArea Height0.88n2 + O(1) 0.66n [Brandenburg 2008]0.44n2 + O(1) 0.66n (polyline) [Bonichon et al. 2003]Planar 3-treesArea Height0.88n2 + O(1) 0.5n [Brandenburg 2008,
Hossain et al. 2013]Nested Triangles GraphArea Height0.22n2 + O(1) 0.33n [Frati and Patrignani 2008]
Upper BoundsVariable Embedding
Improved Upper Bounds
Idea: Use the Simple Cycle Separator
A separator of size O(√n)
An n-vertex planar graph G A simple cycle separator of G
Go with 2n/3+O(√n) vertices Gi with 2n/3+O(√n) vertices
[Djidjev and Venkatesan, 1997] Every planar triangulation has a simple cycle separator of size O(√n)
The Big Picture
w w'
v4
v7
v6v1
v5
v4
v5
v1
v8
v6
v2
v3
Towards w’
a
v8 w b
Gi Go
Choose an Embedding
Decomposition
Drawing and Merge
8
Technical Details (Choose an Embedding)
Choose a face which is incident to some edge of the cycle separator as the
new outer face.
GD 2014September 26,
2014
Technical Details (Construct Go and Gi)
Construct Go and Gi
Gi Go
Choose a face which is incident to some edge of the cycle separator as the
new outer face.
GD 2014September 26,
2014
10
Technical Details (Draw Go and Gi)
Gi Go
GD 2014September 26,
2014
w
11
Technical Details (Draw Go and Gi)
Gi Go
v4
v8 w
wv7
v6
v1
v5
GD 2014September 26,
2014
12
Technical Details (Draw Go and Gi)
Gi Go
v4
v8 w
wv7
v6
v1
v5
w'
GD 2014September 26,
2014
Technical Details (Draw Go and Gi)
GiGo
v4
v8 w
w
v7
v6
v1
v5
Di
w’
v4
v5
v1
v8
v6
v2
v3
Towards w’
Do
b
a
Technical Details (Merge Do and Di)
v4
v8
v7
v6
v1
v5
Di
Do
v2
v3
Technical Details (Merge Do and Di)
v4
v8
v7
v6
v1
v5
Di
Do
v2
v3
Height of Di is (2/3)×|Di| = 4n/9+O(λ)
Height of Do is (2/3)×|Do| = 4n/9+O(λ∆)
Height of the final drawing is 4n/9+O(λ∆)
At most 6 bends per edge - two bends to enter Do from Di
- two bends on separator- two bends to return to Di from Do
Improve to 4 bends per edge using the transformation via
visibility representation [Biedl 2014]
a
b c
d
f e
g
a
b c
a
b c
f
a
b c
d
fa
b c
d
f e
Plane 3-Trees
A planar 3-tree
Start with a triangle, then repeatedly add a vertex and triangulate the resulting graph.
GD 2014 September 26, 2014
Plane 3-Trees
f f
d
f
de
fd
eg
The representative tree
a
b c
d
f e
g
a
b c
a
b c
f
a
b c
d
fa
b c
d
f eA planar 3-tree
Plane 3-Treesr
The representative tree T of G
a
cA planar 3-tree G
b
18GD 2014September 26,
2014
a
c
Plane 3-Trees
A planar 3-tree G
r
v
The representative tree T of G b
v
r
Each component with at most n/2
vertices
a
cA planar 3-tree G
b
v w
Choosing a suitable embedding
v wG1
G3
G2F
F
a
c
Plane 3-Trees
A planar 3-tree Gb
v w
Plane 3 trees inside each of these triangles has n/2+O(1) vertices
v wG1
G3
G2
w1w2
w3
v
w
w1
w2
F
F
x
y
wt
y
x
4n/9 + O(1)
21GD 2014September 26,
2014
a
c
Plane 3-Trees
A planar 3-tree Gb
v w
Choosing a suitable embedding
v wG1
G3
G2
w1w2
w3
v
w
F
F
x
y
wt
y
x
4n/9 + O(1)
v
w
w1
w2
The main challenge here is to show that the number of lines that are intersecting each triangle is sufficient to
draw the corresponding plane 3-tree
22GD 2014September 26,
2014
v4
v8
v7
v6
v1v5
v2
v3
TriangulationsPolyline drawing with height 4n/9+O(λ∆) ≈ 0.44n+O(λ∆)This is 0.44n+o(n) when ∆ is o(n)
Planar 3-treesStraight-line drawing with height 4n/9+O(1) ≈ 0.44n+O(1)
TriangulationsArea Height0.88n2 + O(1) 0.66n [Brandenburg 2008]0.44n2 + O(1) 0.66n (polyline) [Bonichon et al. 2003]
Planar 3-treesArea Height0.88n2 + O(1) 0.5n [Brandenburg 2008,
Hossain et al. 2013]
Upper Bounds Improved Upper Bounds
a
c
r
v
b
v
r
Thank
you
OPEN: Clo
se th
e gap
!