Download G - Tohoku University

Victoria city and Sendai city
7300km
Victoria city
Sendai city
1
2
Tohoku University
Tohoku University was established in 1907.
Spring
Summer
Autumu
Winter
GSIS, Tohoku University
3
Graduate School of Information Sciences (GSIS),
Tohoku University, was established in 1993.
150 Faculties
450 students
Math.
Computer Science
Robotics
Transportation
Economics
Human Social
Sciences
Interdisciplinary
School
Book
4
5
Small Grid Drawings of Planar
Graphs with Balanced Bipartition
Xiao Zhou Takashi Hikino Takao Nishizeki
Graduate School of Information Sciences,
Tohoku University, Japan
6
Grid drawing
In a grid drawing of a planar graph,
・ every vertex is located at a grid point,
・ every edge is drawn as a straight-line segment
without any edge-intersection.
Planar graph
Grid drawing
2
3
2
3
4
4
1
1
5
7
5
6
7
6
7
Embedding
We deal with grid drawings of a planar
graph in variable embedding setting.
1
3
6
7
Planar graph
Grid drawing
2
2
3
4
4
1
1
5
7
4
5
This embedding is different
from a given embedding
3
2
5
6
7
6
Width and Height of grid drawing
H
H
W
W
W=9, H=11
W=4, H=3
Area W×H=99
Area W×H=12
8
9
Small grid drawing
Large area
Small area
H
H
W
W
We wish to find a small grid drawing
in variable embedding setting.
10
Known results
Grid drawing
of Plane graph
Width and Height
Running time
[Schnyder, 1990]
[Chrobak, Kant, 1997]
W=n－2, H=n－2
O(n)
n ： number of vertices
11
Our results
Grid drawing
of Plane graph
Width and Height
Running time
[Schnyder, 1990]
[Chrobak, Kant, 1997]
W=n－2, H=n－2
O(n)
If a planar graph G has a balanced bipartition,
then G has a grid drawing with small area.
s
G
s
G2
G1
G2
s
t
Planar graph G
G1
G2
t Drawing
Bipartition
t
Subgraph G1,G2
u2
s
u1
G1
t
Drawing of G
12
Outline of algorithm
s
G
s
G2
G1
G2
s
Bipartition
t
t
G1
(1) Planar graph G
t
s
(2) Subgraph G1,G2
G1
u1
s
G2
u2
t
t
(3) Maximal planar graph G1,G2
u2
s
G2
u1
G1
s
G2
Combining
t
(5) Drawing of G
u2
s
G1
u1
t
t
(4) Drawing of G1,G2
13
Our results
s
G
G2
G1
Theorem 1
t
(1) Planar graph G
If a planar graph G has a balanced
bipartition, then G has a grid
drawing with small area.
W≤max{n1, n2}－1
u2
s
G2
u1
G1
H≤max{n1, n2}－2
t
(5) Drawing of G
G1,G2
Width, Height
n1,n2≤n/2
W,H≤n/2
n1,n2≤2n/3
W,H≤2n/3
n1,n2≤αn
W,H≤αn
If α<1, Balanced bipartition
14
Our results
Theorem 1
If a planar graph G has a balanced
bipartition, then G has a grid
drawing with small area.
This graph has NO
balanced bipartition.
G1,G2
Width, Height
n1,n2≤n/2
W,H≤n/2
n1,n2≤2n/3
W,H≤2n/3
n1,n2≤αn
W,H≤αn
If α<1, Balanced bipartition
15
Our results
Theorem 1
If a planar graph G has a balanced
bipartition, then G has a grid
drawing with small area.
This graph has NO
balanced bipartition.
Planar graph
Series-Parallel
graph
Lemma 1
Every Series-Parallel graph has a
balanced bipartition (n1,n2≤2n/3).
α=2/3
16
Our results
Theorem 1
If a planar graph G has a balanced
bipartition, then G has a grid
drawing with small area.
This graph has NO
balanced bipartition.
Planar graph
Series-Parallel
graph
Lemma 1
Every Series-Parallel graph has a
balanced bipartition (n1,n2≤2n/3).
α=2/3
Theorem 2
Series-Parallel W   2 n ,   2 n   1
H  


graph
3 
3 
17
Outline of algorithm
s
G
s
G2
G1
G2
s
Bipartition
t
t
G1
(1) Planar graph G
t
s
(2) Subgraph G1,G2
G1
u1
s
G2
u2
t
t
(3) Maximal planar graph G1,G2
u2
s
G2
u1
G1
s
G2
Combining
t
(5) Drawing of G
u2
s
G1
u1
t
t
(4) Drawing of G1,G2
18
Bipartition
We call a pair of distinct vertices {s,t} in a graph G=(V,E) a
separation pair of G if G has two subgraphs G1=(V1,E1) and
G2=(V2,E2) such that
・ V=V1∪V2，V1∩V2={s,t},
・ E=E1∪E2，E1∩E2=∅．
Such a pair of subgraphs {G1,G2} is called a bipartition of G.
s
s
G2
s
n1=9
Bipartition
t
t
G1
n=12
(1) Graph G
(2) Subgraph G1,G2
t
n2=5
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Bipartition
We call a pair of distinct vertices {s,t} in a graph G=(V,E) a
separation pair of G if G has two subgraphs G1=(V1,E1) and
G2=(V2,E2) such that
・ V=V1∪V2，V1∩V2={s,t},
・ E=E1∪E2，E1∩E2=∅．
Such a pair of subgraphs {G1,G2} is called a bipartition of G.
t
t
t
G2
G1
Bipartition
s
n=12
s
s
n1=3
(1) Graph G
n2=11
(2) Subgraph G1,G2
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Outline of algorithm
s
G
s
G2
G1
G2
s
Bipartition
t
t
G1
(1) Planar graph G
t
s
(2) Subgraph G1,G2
G1
u1
s
G2
u2
t
t
(3) Maximal planar graph G1,G2
u2
s
G2
u1
G1
s
G2
Combining
t
(5) Drawing of G
u2
s
G1
u1
t
t
(4) Drawing of G1,G2
21
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
W≤max{n1, n2}－1
s
G
u2
s
G2
G1
t
Planar graph G
Drawing
in linear time
G2
u1
G1
t
Drawing of G
H≤max{n1, n2}－2
22
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Assume w.l.o.g. that n1≥n2．
s
s
s
G1
t
G
n=12
t
t
G1
n1=9
n2=5
G2
23
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Assume w.l.o.g. that n1≥n2．
Add dummy edges to G1 so that the resulting graph is maximal planar
and has an edge (s,t).
s
s
G1
t
G
n=12
t
G1
n1=9
24
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Embed G1 so that the edge (s,t) lies on the outer face of G1.
s
s
G1
t
G
n=12
u1
G1
t
n1=9
25
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Obtain a grid drawing of G1[CK97].
s
s
G1
t
G
n=12
u1
G1
t
n1=9
26
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
s
bipartition of G. Then G has a grid drawing such that W≤max{n
1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Obtain a grid drawing of G1[CK97].
u1
G1
s
H1=n1－2
Edge (u1,t) is horizontal.
G1
n1=9
u1
t
W1=n1－2
t
n1=9
27
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
u2
s
s
G2
s
G2
t
t
u1
G
n=12
t
G1
n1=9
n2=5
G2
28
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Add n1－n2 dummy vertices to G2 so that the resulting graph has
exactly n1 vertices.
u2
s
s
G2
s
G2
t
t
u1
G
n=12
t
G1
n1=9
n2=5
=n1=9 G2
29
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Add dummy edges to G2 so that the resulting graph is maximal planar
and has an edge (s,t).
s
G2
t
G
n=12
s
t
n2=n1=9 G2
30
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Embed G2 so that the edge (s,t) lies on the outer face of G2.
s
G2
t
G
n=12
s
t
n2=n1=9 G2
31
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Embed G2 so that the edge (s,t) lies on the outer face of G2.
s
G2
t
G
n=12
u2
s
s
t
t
n2=n1=9 G2
n2=n1=9 G2
32
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Obtain a grid drawing of G2[CK97].
Edge (u2,s) is horizontal.
s
G2
t
t
u2
s
t
n2=n1=9 G2
G
n=12
u2
s
Theorem 1
s
u1
t
s
u1
u2
s
G1
G2
n1=9
n2=9
t
u2
s
t
33
t
Theorem 1
G
s
Combine the two drawings
and Erase all the dummy
vertices and edges.
u2
u1
34
t
n=12
s
u2
s
G2
n2=9
G1
n1=9
u1
t
t
Theorem 1
G
s
Combine the two drawings
and Erase all the dummy
vertices and edges.
u2
u1
35
t
n=12
u2
s
G2
n2=9
G1
n1=9
u1
t
36
Theorem 1
Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary
bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1,
H≤max{n1,n2}－2 and such a drawing can be found in linear time.
Such a drawing can be found in linear time, because drawings of G1,G2
can be found in linear time by the algorithm in CK97.
W2=n1－2
n1≥n2
u2
s
G2
s
H1=n1－2
u1
t
G1
t
W1=n1－2
u2
s
G2
H2=n1－2
u1
G1
t
W=W1+1
=n1－1
=max{n1, n2}－1
H=H1
=n1－2
=max{n1, n2}－2
Q.E.D.
Our results
37
Theorem 1
If a planar graph G has a balanced
bipartition, then G has a grid
drawing with small area.
This graph has NO
balanced bipartition.
Lemma 1
Every Series-Parallel graph has a
balanced bipartition (n1,n2≤2n/3).
Planar graph
Series-Parallel
graph
Theorem 2
Series-Parallel W   2 n ,   2 n   1
H  


graph
3 
3 
Our results
38
Theorem 1
If a planar graph G has a balanced
bipartition, then G has a grid
drawing with small area.
This graph has NO
balanced bipartition.
Lemma 1
Every Series-Parallel graph has a
balanced bipartition (n1,n2≤2n/3).
Planar graph
Series-Parallel
graph
Theorem 2
Series-Parallel W   2 n ,   2 n   1
H  


graph
3 
3 
39
Series-Parallel graph
A Series-Parallel graph is recursively defined as follows:
(1)
(2)
A single edge is a SP graph．
G2
G1
terminal
: SP graph
Series
connection
G1
SP graph
G2
G1
G2
Parallel
connection
SP graph
40
Series-Parallel graphs
These graphs are Series-Parallel.
s
t
Bipartition of Series-Parallel graph
41
Lemma 1. Every Series-Parallel graph G of n vertices has
a bipartition {G1,G2} such that n1,n2  2n / 3  1 .
Furthermore such a bipartition can be found in linear time.
s
G
G2
G1
t
SP graph G
Bipartition
in liner time
s
s
G2
G1
t
t
Subgraph G1,G2
n1,n2  2n / 3  1
Suppose for a contradiction that a SP graph has no desired bipartition.
Bipartition of Series-Parallel graph
Lemma 1. Every Series-Parallel graph G of n vertices has
a bipartition {G1,G2} such that n1,n2  2n / 3  1 .
Furthermore such a bipartition can be found in linear time.
Let {s,t} be the most balanced separation pair of G :
max{n1,n2} is minimum among all bipartitions of G.
n1>2n/3
G1
G11
s
G1=G11・G12
G11
G12
u G12
t
G2
n2<n/3
2-connected SP graph G
Assume w.l.o.g. that
n1≥n2．
42
Bipartition of Series-Parallel graph
43
Lemma 1. Every Series-Parallel graph G of n vertices has
a bipartition {G1,G2} such that n1,n2  2n / 3  1 .
Furthermore such a bipartition can be found in linear time.
Let {s,t} be the most balanced separation pair of G :
max{n1,n2} is minimum among all bipartitions of G.
n1>2n/3
G1=G11・G12 Assume w.l.o.g. that
n1≥n2 and n11≥n12.
G1 n11>n/3
G11
n1> n11
u G12
n1> n11
G
s
t
11
n1> max{n11,n11}
G2
Contradiction.
n11<2n/3
n1=max{n1,n2}
n2<n/3
max{n1,n2}> max{n11,n11}
2-connected SP graph G
Grid drawing of Series-Parallel graph
s
G
Lemma 1
in linear time
G2
G1
t
SP graph G
s
s
G2
G1
t
t
Subgraph G1,G2
n1,n2  2n / 3  1
44
Grid drawing of Series-Parallel graph
s
G
G1
SP graph G
u2
s
G2
t
t
t
Subgraph G1,G2
n1,n2  2n / 3  1
s
G1
G2
G1
t
s
s
s
Lemma 1
in linear time
G2
45
t
Subgraph G1,G2
Theorem 1
in linear time
G2
u1
H≤max{n1, n2}－2
G1
t
W≤max{n1, n2}－1
Grid drawing of Series-Parallel graph
s
G
G1
SP graph G
G1
t
t
Subgraph G1,G2
n1,n2  2n / 3  1
u2
s
s
G2
t
G2
G1
t
s
s
s
Lemma 1
in linear time
G2
46
t
SP Subgraph G1,G2
n1,n2  2n / 3  1
Theorem 1
in linear time
G2
u1
G1
H≤max{n1, n2}－2
 2n / 3  1
t
W≤max{n1, n2}－1
 2n / 3
Grid drawing of Series-Parallel graph
47
Theorem 2. Every Series-Parallel graph of n vertices has
a grid drawing such that W  2n / 3 , H  2n / 3  1 .
Furthermore such a drawing can be found in linear time.
s
G1
G2
t
u2
s
s
t
SP Subgraph G1,G2
n1,n2  2n / 3  1
Theorem 1
in linear time
G2
u1
G1
H≤max{n1, n2}－2
 2n / 3  1
t
W≤max{n1, n2}－1
 2n / 3
48
Grid drawing of Series-Parallel graph
u2
s
s
H=7
Theorem 2
in linear time
t
n=12
SP graph G
u1
t
W=8
s
s
G1
n1=9
t
G2
t
n2=5
49
Conclusions
Gird drawing
Width and Height
Running time
W≤max{n1, n2}－1
Planar graph
with balanced
bipartition
u2
s
G2
u1
O(n)
H≤max{n1, n2}－2
G1
t
2 
u2
s
SP graph
Partial 2-tree
G2
u1
G1
t
W   n
3 
2 
H   n  1
3 
O(n)
50