Victoria city and Sendai city

1
Victoria city and Sendai city
7300km
Victoria city
Sendai city
2
Tohoku University
Tohoku University was established in 1907.
Spring
Summer
Autumu
Winter
3
GSIS, Tohoku University
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
4
Book
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
2
3
3
4
4
1
1
5
5
6
6
7
7
7
Embedding
We deal with grid drawings of a planar graph in
variable embedding setting.
2
1
4
3
6
5
This embedding is different from a given embedding
7
Planar graph
Grid drawing
2
2
3
3
4
4
1
1
5
5
6
6
7
7
8
Width and Height of grid drawing
H
H
W
W
W9, H11
W4, H3
Area WH99
Area WH12
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 Wn-2, Hn-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 Wn-2, Hn-2 O(n)
If a planar graph G has a balanced bipartition,
then G has a grid drawing with small area.
s
s
Bipartition
Drawing
G2
s
G2
s
G2
G1
t
G1
G1
t
G
t
t
Planar graph G
Subgraph G1,G2
Drawing of G
12
Outline of algorithm
Maximal planar graph
s
Bipartition
s
G2
s
G2
t
G1
G1
s
t
G
G2
t
s
t
(1) Planar graph G
(2) Subgraph G1,G2
G1
t
(3) Maximal planar graph G1,G2
Drawing
Combining
s
s
s
G2
G2
G1
G1
t
t
t
(4) Drawing of G1,G2
(5) Drawing of G
13
Our results
s
G2
Theorem 1
G1
If a planar graph G has a balanced bipartition,
then G has a grid drawing with small area.
t
G
(1) Planar graph G
G1,G2 Width, Height
n1,n2n/2 W,Hn/2


Wmaxn1, n2-1
s
n1,n22n/3
W,H2n/3
G2
Hmaxn1, n2-2
G1
W,Han
n1,n2an
t
If alt1, Balanced bipartition
(5) Drawing of G
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,n2n/2 W,Hn/2


n1,n22n/3
W,H2n/3
W,Han
n1,n2an
If alt1, 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.
Lemma 1
Every Series-Parallel graph has a balanced
bipartition (n1,n22n/3).
Planar graph
a2/3
Series-Parallel graph
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.
Lemma 1
Every Series-Parallel graph has a balanced
bipartition (n1,n22n/3).
Planar graph
a2/3
Series-Parallel graph
Theorem 2
Series-Parallel graph
W
H
17
Outline of algorithm
Maximal planar graph
s
Bipartition
s
G2
s
G2
t
G1
G1
s
t
G
G2
t
s
t
(1) Planar graph G
(2) Subgraph G1,G2
G1
t
(3) Maximal planar graph G1,G2
Drawing
Combining
s
s
s
G2
G2
G1
G1
t
t
t
(4) Drawing of G1,G2
(5) Drawing of G
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
VV1?V2,V1nV2s,t, EE1?E2,E1nE2Ø. Such a
pair of subgraphs G1,G2 is called a bipartition
of G.
G2
s
s
s
Bipartition
n19
t
t
t
G1
n25
n12
(1) Graph G
(2) Subgraph G1,G2
19
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
VV1?V2,V1nV2s,t, EE1?E2,E1nE2Ø. Such a
pair of subgraphs G1,G2 is called a bipartition
of G.
G2
t
t
t
G1
Bipartition
s
s
s
n13
n12
n211
(1) Graph G
(2) Subgraph G1,G2
20
Outline of algorithm
Maximal planar graph
s
Bipartition
s
G2
s
G2
t
G1
G1
s
t
G
G2
t
s
t
(1) Planar graph G
(2) Subgraph G1,G2
G1
t
(3) Maximal planar graph G1,G2
Drawing
Combining
s
s
s
G2
G2
G1
G1
t
t
t
(4) Drawing of G1,G2
(5) Drawing of G
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 Wmaxn1,n2-1, Hmax
n1,n2-2 and such a drawing can be found in
linear time.
Wmaxn1, n2-1
Drawing in linear time
s
s
G2
G2
Hmaxn1, n2-2
G1
G1
t
G
t
Planar graph G
Drawing of G
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 Wmaxn1,n2-1, Hmax
n1,n2-2 and such a drawing can be found in
linear time.
Assume w.l.o.g. that n1n2.
s
s
s
G1
t
t
t
G2
n25
G1
n19
G
n12
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 Wmaxn1,n2-1, Hmax
n1,n2-2 and such a drawing can be found in
linear time.
Assume w.l.o.g. that n1n2.
Add dummy edges to G1 so that the resulting graph
is maximal planar and has an edge (s,t).
s
s
G1
t
t
G1
n19
G
n12
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 Wmaxn1,n2-1, Hmax
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
t
G1
n19
G
n12
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 Wmaxn1,n2-1, Hmax
n1,n2-2 and such a drawing can be found in
linear time.
Obtain a grid drawing of G1CK97.
s
s
G1
t
t
G1
n19
G
n12
26
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 Wmaxn1,n2-1, Hmax
n1,n2-2 and such a drawing can be found in
linear time.
s
Obtain a grid drawing of G1CK97.
t
G1
s
n19
H1n1-2
Edge (u1,t) is horizontal.
G1
n19
t
W1n1-2
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 Wmaxn1,n2-1, Hmax
n1,n2-2 and such a drawing can be found in
linear time.
s
G2
s
s
G2
t
t
G2
n25
t
G
n12
G1
n19
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 Wmaxn1,n2-1, Hmax
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.
s
G2
s
s
G2
t
t
G2
n2n19
n25
t
G
n12
G1
n19
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 Wmaxn1,n2-1, Hmax
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).
G2
s
s
t
t
G2
n2n19
G
n12
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 Wmaxn1,n2-1, Hmax
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.
G2
s
s
t
t
G2
n2n19
G
n12
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 Wmaxn1,n2-1, Hmax
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.
G2
s
s
s
t
t
t
G2
G2
n2n19
n2n19
G
n12
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 Wmaxn1,n2-1, Hmax
n1,n2-2 and such a drawing can be found in
linear time.
Obtain a grid drawing of G2CK97.
Edge (u2,s) is horizontal.
G2
s
s
t
t
G2
n2n19
G
n12
33
Theorem 1
s
s
t
t
G1
G2
n19
n29
u2
s
s
u1
t
t
34
Theorem 1
G
s
Combine the two drawings and Erase all the dummy
vertices and edges.
t
n12
s
G2
n29
G1
n19
t
35
Theorem 1
G
s
Combine the two drawings and Erase all the dummy
vertices and edges.
t
n12
s
G2
n29
G1
n19
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 Wmaxn1,n2-1, Hmax
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.
W2n1-2
n1n2
s
s
G2
HH1 n1-2 maxn1, n2-2
G2
G1
H2n1-2
s
t
t
H1n1-2
G1
t
WW11 n1-1 maxn1, n2-1
Q.E.D.
W1n1-2
37
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.
Lemma 1
Every Series-Parallel graph has a balanced
bipartition (n1,n22n/3).
Planar graph
Series-Parallel graph
Theorem 2
Series-Parallel graph
W
H
38
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.
Lemma 1
Every Series-Parallel graph has a balanced
bipartition (n1,n22n/3).
Planar graph
Series-Parallel graph
Theorem 2
Series-Parallel graph
W
H
39
Series-Parallel graph
A Series-Parallel graph is recursively defined as
follows
(1)
terminal
A single edge is a SP graph.
(2)
(2)
SP graph
G2
G2
G1
G1
Series connection
SP graph
G1
G2
G1
Parallel connection
SP graph
G2
40
Series-Parallel graphs
These graphs are Series-Parallel.
s
t
41
Bipartition of Series-Parallel graph
Lemma 1. Every Series-Parallel graph G of n
vertices has a bipartition G1,G2 such that
n1,n2 . Furthermore such a
bipartition can be found in linear time.
Bipartition in liner time
s
s
s
G2
G2
t
G1
G1
t
t
G
SP graph G
Subgraph G1,G2
n1,n2
Suppose for a contradiction that a SP graph has
no desired bipartition.
42
Bipartition of Series-Parallel graph
Lemma 1. Every Series-Parallel graph G of n
vertices has a bipartition G1,G2 such that
n1,n2 . Furthermore such a
bipartition can be found in linear time.
Let s,t be the most balanced separation pair of
G maxn1,n2 is minimum among all bipartitions
of G.
n1gt2n/3
Assume w.l.o.g. that n1n2.
G1G11G12
G1
G11
G11
u
G12
G12
s
t
G2
n2ltn/3
SP graph G
2-connected
43
Bipartition of Series-Parallel graph
Lemma 1. Every Series-Parallel graph G of n
vertices has a bipartition G1,G2 such that
n1,n2 . Furthermore such a
bipartition can be found in linear time.
Let s,t be the most balanced separation pair of
G maxn1,n2 is minimum among all bipartitions
of G.
n1gt2n/3
Assume w.l.o.g. that n1n2 and n11n12.
G1G11G12
n11gtn/3
G1
G11
n1gt n11
u
G12
n1gt n11
G11
s
t
n1gt maxn11,n11
G2
Contradiction.
n11lt2n/3
n1maxn1,n2
n2ltn/3
maxn1,n2gt maxn11,n11
SP graph G
2-connected
44
Grid drawing of Series-Parallel graph
s
Lemma 1 in linear time
s
s
G2
G2
G1
G1
t
t
t
G
SP graph G
Subgraph G1,G2
n1,n2
45
Grid drawing of Series-Parallel graph
s
Lemma 1 in linear time
s
s
G2
G2
G1
G1
t
t
t
G
SP graph G
Subgraph G1,G2
n1,n2
s
s
Theorem 1 in linear time
G2
s
G2
Hmaxn1, n2-2
G1
G1
t
t
t
Subgraph G1,G2
Wmaxn1, n2-1
46
Grid drawing of Series-Parallel graph
s
Lemma 1 in linear time
s
s
G2
G2
G1
G1
t
t
t
G
SP graph G
Subgraph G1,G2
n1,n2
s
s
Theorem 1 in linear time
G2
s
G2
Hmaxn1, n2-2
G1
G1
t
t
t
SP
Subgraph G1,G2
Wmaxn1, n2-1
n1,n2
47
Grid drawing of Series-Parallel graph
Theorem 2. Every Series-Parallel graph of n
vertices has a grid drawing such that W
H . Furthermore
such a drawing can be found in linear time.
s
s
Theorem 1 in linear time
G2
s
G2
Hmaxn1, n2-2
G1
G1
t
t
t
SP
Subgraph G1,G2
Wmaxn1, n2-1
n1,n2
48
Grid drawing of Series-Parallel graph
s
s
Theorem 2 in linear time
H7
t
SP graph G
n12
t
W8
s
Lemma 1 in linear time
Theorem 1 in linear time
s
G1
n19
G2
t
t
n25
49
Conclusions
Gird drawing Width and Height Running time
Planar graph with balanced bipartition O(n)
SP graph Partial 2-tree O(n)
Wmaxn1, n2-1
s
G2
Hmaxn1, n2-2
G1
t
W
s
G2
G1
H
t
50
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51
Example of Series-Parallel graph
e2
e3
e1
e6
e4
e7
e5
SP graph
52
Example of Series-Parallel graph
A single edge is a SP graph.
e2
e2
e3
e1
e3
e1
e6
e6
e4
e7
e7
e4
e5
e5
terminal
SP graph
53
Example of Series-Parallel graph
Series connection
SP graph
G1
G2
e2
e2
e3
e1
e3
e1
e6
e6
e4
e7
e7
e4
e5
e5
terminal
SP graph
54
Example of Series-Parallel graph
Series connection
SP graph
G1
G2
e2
e2
e3
e1
e3
e1
e6
e6
e4
e7
e7
e4
e5
e5
terminal
SP graph
55
Example of Series-Parallel graph
Parallel connection
G1
SP graph
G2
e2
e2
e3
e1
e3
e1
e6
e6
e4
e7
e7
e4
e5
e5
terminal
SP graph
56
Example of Series-Parallel graph
Series connection
SP graph
G1
G2
e2
e2
e3
e1
e3
e1
e6
e4
e7
e6
e7
e4
e5
e5
terminal
SP graph
57
Example of Series-Parallel graph
Series connection
SP graph
G1
G2
e2
e2
e3
e1
e3
e1
e6
e4
e7
e6
e7
e4
e5
e5
terminal
SP graph
58
Example of Series-Parallel graph
Parallel connection
G1
SP graph
G2
e2
e2
e3
e1
e3
e1
e6
e4
e7
e6
e4
e7
e5
e5
terminal
SP graph
59
Example of Series-Parallel graph
Parallel connection
G1
SP graph
G2
e2
e2
e3
e1
e3
e1
e6
e6
e4
e7
e4
e7
e5
e5
terminal
SP graph