Drawing Plane Graphs

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Drawing Plane Graphs
Takao Nishizeki
Tohoku University
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(No Transcript)
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US President
California Governor
What is the common feature?
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Graphs and Graph Drawings
A diagram of a computer network
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Objectives of Graph Drawings
Nice drawing
structure of the graph is easy to understand
structure of the graph is difficult to understand
To obtain a nice representation of a graph so
that the structure of the graph is easily
understandable.
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Objectives of Graph Drawings
Nice drawing
Ancient beauty
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Objectives of Graph Drawings
Diagram of an electronic circuit
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Wire crossings
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4
8
3
1
2
not suitable for single layered PCB
suitable for single layered PCB
The drawing should satisfy some criterion arising
from the application point of view.
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Drawings of Plane Graphs
Straight line drawing
Convex drawing
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Drawings of Plane Graphs
Rectangular drawing
Box-rectangular drawing
Orthogonal drawing
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Book
Planar Graph Drawing by Takao Nishizeki Md.
Saidur Rahman
http//www.nishizeki.ecei.tohoku.ac.jp/nszk/saidur
/gdbook.html
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Straight Line Drawing
Plane graph
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Straight Line Drawing
Straight line drawing
Plane graph
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Straight Line Drawing
Straight line drawing
Plane graph
Each vertex is drawn as a point.
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Straight Line Drawing
Straight line drawing
Plane graph
Each vertex is drawn as a point.
Each edge is drawn as a single straight line
segment.
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Every plane graph has a straight line
drawing. Wagner 36 Fary 48
Straight Line Drawing
Polynomial-time algorithm
Straight line drawing
Plane graph
Each vertex is drawn as a point.
Each edge is drawn as a single straight line
segment.
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Straight Line Drawing
W
H
Straight line drawing
Plane graph
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Straight Line Grid Drawing
Straight line grid drawing.
Plane graph
In a straight line grid drawing each vertex is
drawn on a grid point.
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Wagner 36 Fary 48
Grid-size is not polynomial of the number of
vertices n
Straight line grid drawing.
Plane graph
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Straight Line Grid Drawing
Straight line grid drawing.
Plane graph
de Fraysseix et al. 90
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Chrobak and Payne 95 Linear algorithm
Straight line grid drawing.
Plane graph
de Fraysseix et al. 90
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Schnyder 90
H
n-2
n-2
W
Upper bound
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What is the minimum size of a grid required for a
straight line drawing?
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Lower Bound
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A restricted class of plane graphs may have more
compact grid drawing.
Triangulated plane graph
3-connected graph
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4-connected ?
disconnected
not 4-connected
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How much area is required for 4-connected plane
graphs?
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Straight line grid drawing
Miura et al. 01
Input 4-connected plane graph G
Output a straight line grid
drawing Grid Size
H lt n/2

Area
W lt n/2

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Miura et al. 01
Schnyder 90
4-connected plane graph G
plane graph G
n-2
H lt n/2

W lt n/2
n-2

.
Arean
Arealtn /4
.
2
2

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The algorithm of Miura et al. is best possible
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Main idea
G
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Draw a graph G on the plane nicely
Straight line drawing
A convex drawing is a straight line drawing where
each face is drawn as a convex polygon.
Convex drawing
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Convex Drawing
Convex drawing
Tutte 1963
Every 3-connected planar graph has a convex
drawing.
A necessary and sufficient condition for a plane
graph to have a convex drawing. Thomassen
80
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Convex Drawing
Chiba et al. 84 O(n) time algorithm
Convex drawing
Tutte 1963
Every 3-connected planar graph has a convex
drawing
A necessary and sufficient condition for a plane
graph to have a convex drawing. Thomassen
80
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Convex Grid Drawing
Chrobak and Kant 97
Input 3-connected graph
Output convex grid drawing
n-2
n-2
Grid Size
Area
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Convex Grid Drawing
Miura et al. 2000
Input 4-connected plane graph
Output
Convex grid drawing
H
Grid Size
W
Half-perimeter
Area
36
Miura et al. 2000
Chrobak and Kant 97
3-connected graph
4-connected graph
n-2
H
n-2
W
Area
Area
37
The algorithm of Miura et al. is best possible
H
W
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Main idea
20
21
19
18
17
14
15
13
10
12
16
11
9
6
8
7
3
4
5
1
2
1 4-canonical decomposition
2 Find paths
O(n)NRN97
3 Decide x-coordinates
Time complexity O(n)
39
VLSI Floorplanning
B
A
F
E
C
G
D
Interconnection graph
40
VLSI Floorplanning
B
B
A
A
F
F
E
E
C
G
C
G
D
D
VLSI floorplan
Interconnection graph
41
VLSI Floorplanning
B
B
A
A
F
F
E
E
C
G
C
G
D
D
VLSI floorplan
Interconnection graph
42
VLSI Floorplanning
B
B
A
A
F
F
E
E
C
G
C
G
D
D
VLSI floorplan
Interconnection graph
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VLSI Floorplanning
B
B
A
A
F
F
E
E
C
G
C
G
D
D
VLSI floorplan
Interconnection graph
Dual-like graph
44
VLSI Floorplanning
B
B
A
A
F
F
E
E
C
G
C
G
D
D
VLSI floorplan
Interconnection graph
Dual-like graph
Add four corners
45
VLSI Floorplanning
B
B
A
A
F
F
E
Rectangular drawing
E
C
G
C
G
D
D
VLSI floorplan
Interconnection graph
Dual-like graph
Add four corners
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Rectangular Drawings
Plane graph G of
Input
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Rectangular Drawings
corner
Rectangular drawing of
G
Plane graph G of
Output
Input
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Rectangular Drawings
corner
Rectangular drawing of
G
Plane graph G of
Output
Input
Each vertex is drawn as a point.
49
Rectangular Drawings
corner
Rectangular drawing of
G
Plane graph G of
Output
Input
Each vertex is drawn as a point.
Each edge is drawn as a horizontal or a vertical
line segment.
50
Rectangular Drawings
corner
Rectangular drawing of
G
Plane graph G of
Output
Input
Each vertex is drawn as a point.
Each edge is drawn as a horizontal or a vertical
line segment.
Each face is drawn as a rectangle.
51
Not every plane graph has a rectangular drawing.
52
Thomassen 84,
Necessary and sufficient condition
Rahman, Nakano and Nishizeki 98
Linear-time algorithms
Rahman, Nakano and Nishizeki 02
Linear algorithm for the case where corners are
not designated in advance
53
A Necessary and Sufficient Condition by Thomassen
84
plane graph
G
four vertices of degree 2 are designated as
corners
G has a rectangular drawing if and only if
every 2-legged cycle in G contains at least two
designated vertices and
every 3-legged cycle in G contains at least one
designated vertex.
2-legged cycles
3-legged cycles
54
A Necessary and Sufficient Condition by Thomassen
84
plane graph
G
four vertices of degree 2 are designated as
corners
G has a rectangular drawing if and only if
every 2-legged cycle in G contains at least two
designated vertices and
every 3-legged cycle in G contains at least one
designated vertex.
2-legged cycles
55
A Necessary and Sufficient Condition by Thomassen
84
plane graph
G
four vertices of degree 2 are designated as
corners
G has a rectangular drawing if and only if
every 2-legged cycle in G contains at least two
designated vertices and
every 3-legged cycle in G contains at least one
designated vertex.
2-legged cycles
3-legged cycles
56
A Necessary and Sufficient Condition by Thomassen
84
plane graph
G
four vertices of degree 2 are designated as
corners
G has a rectangular drawing if and only if
every 2-legged cycle in G contains at least two
designated vertices and
every 3-legged cycle in G contains at least one
designated vertex.
Bad cycles
2-legged cycles
3-legged cycles
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Outline of the Algorithm of Rahman et al.
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Outline of the Algorithm of Rahman et al.
59
Outline of the Algorithm of Rahman et al.
60
Outline of the Algorithm of Rahman et al.
61
Outline of the Algorithm of Rahman et al.
62
Outline of the Algorithm of Rahman et al.
63
(No Transcript)
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Bad cycle
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Bad cycle
Bad cycle
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Partition-pair
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Splitting
Partition-pair
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Partition-pair
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Partition-pair
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Partition-pair
Rectangular drawing
71
Time complexity O(n)
Partition-pair
Rectangular drawing
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Miura, Haga, N. 03, Working paper
Rectangular drawing of plane graph G with ? 4
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G
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G
Gd
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G
Gd
perfect matching
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G
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VLSI Floorplanning
B
B
A
A
F
F
E
Rectangular drawing
E
C
G
C
G
D
D
VLSI floorplan
Interconnection graph
78
VLSI Floorplanning
B
B
A
A
F
F
E
Rectangular drawing
E
C
G
C
G
D
D
VLSI floorplan
Interconnection graph
Unwanted adjacency
Not desirable for MCM floorplanning and for some
architectural floorplanning.
79
B
B
A
F
A
F
E
G
C
E
C
G
D
D
MCM Floorplanning
Sherwani Architectural Floorplanning
Munemoto, Katoh, Imamura
Interconnection graph
80
B
B
A
F
A
F
E
G
C
E
C
G
D
D
MCM Floorplanning Architectural Floorplanning
Interconnection graph
81
B
B
A
F
A
F
E
G
C
E
C
G
D
D
MCM Floorplanning Architectural Floorplanning
Interconnection graph
Dual-like graph
82
B
B
A
F
A
F
E
G
C
E
C
G
D
D
MCM Floorplanning Architectural Floorplanning
Interconnection graph
Dual-like graph
83
B
B
A
F
A
F
E
G
C
E
C
G
D
D
MCM Floorplanning Architectural Floorplanning
Interconnection graph
Dual-like graph
84
Box-Rectangular drawing
B
B
A
F
A
F
E
dead space
G
C
E
C
G
D
D
MCM Floorplanning Architectural Floorplanning
Interconnection graph
Dual-like graph
85
Box-Rectangular Drawing
b
a
c
d
Plane multigraph G
Input
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Box-Rectangular Drawing
a
b
b
a
d
c
c
d
Box-rectangular drawing
Plane multigraph G
Input
Output
87
Box-Rectangular Drawing
a
b
b
a
d
c
c
d
Box-rectangular drawing
Plane multigraph G
Input
Output

• Each vertex is drawn as a rectangle.

88
Box-Rectangular Drawing
a
b
b
a
d
c
c
d
Box-rectangular drawing
Plane multigraph G
Input
Output

• Each vertex is drawn as a rectangle.
• Each edge is drawn as a horizontal or a
  vertical line segment.

89
Box-Rectangular Drawing
a
b
b
a
d
c
c
d
Box-rectangular drawing
Plane multigraph G
Input
Output

• Each vertex is drawn as a rectangle.
• Each edge is drawn as a horizontal or a
  vertical line segment.
• Each face is drawn as a rectangle.

90
Rahman et al. 2000
A necessary and sufficient condition for a plane
multigraph to have a box-rectangular drawing.
91
Algorithm of Rahman et al.
Main Idea Reduction to a rectangular
drawing problem
92
Outline
G
93
Replace each vertex of degree four or more by a
cycle
Outline
G
94
Outline
G
linear time
RNN98a
Rectangular drawing
95
Outline
G
linear time
RNN98a
Box-rectangular drawing
96
Outline
Time complexity O(n)
G
linear time
RNN98a
Box-rectangular drawing
97
Orthogonal Drawings
f
g
e
h
d
i
c
b
a
plane graph G
Input
98
Orthogonal Drawings
i
f
g
h
e
g
h
d
f
e
d
i
c
c
b
a
b
a
plane graph G
Output
Input
99
Orthogonal Drawings
i
f
g
h
e
g
h
bends
d
f
e
d
i
c
c
b
a
b
a
plane graph G
Output
Input
Each edge is drawn as an alternating sequence of
horizontal and vertical line segments.
100
Orthogonal Drawings
i
f
g
h
e
g
bend
h
d
f
e
d
i
c
c
b
a
b
a
plane graph G
Output
Input
Each edge is drawn as an alternating sequence of
horizontal and vertical line segments.
Each vertex is drawn as a point.
101
Applications
Entity-relationship diagrams
Flow diagrams
102
Applications
Circuit schematics
Minimization of bends reduces the number of
vias or throughholes, and hence reduces VLSI
fabrication costs.
103
Objective
i
f
g
h
e
g
h
9 bend
d
f
e
d
i
c
c
b
a
b
a
plane graph G
g
f
4 bends
h
To minimize the number of bends in an orthogonal
drawing.
e
i
d
a
c
b
minimum number of bends.
104
Bend-Minimum Orthogonal Drawing
Garg and Tamassia 96
plane graph of ? 4
time for
Idea
reduction to a minimum cost flow problem
Rahman, Nakano and Nishizeki 99
Linear
for 3-connected cubic plane graph
Idea
reduction to a rectangular drawing problem.
Rahman and Nishizeki 02
Linear
for plane graph of ?3
105
Outline of the algorithm of RNN99
106
Open Problems and Future Research Direction
More area-efficient straight-line or convex
drawing algorithms.
Prescribed face areas
Prescribed length of some edges
Prescribed positions of some vertices
107
Book
Planar Graph Drawing by Takao Nishizeki Md.
Saidur Rahman
http//www.nishizeki.ecei.tohoku.ac.jp/nszk/saidur
/gdbook.html
108
Book
Planar Graph Drawing by Takao Nishizeki Md.
Saidur Rahman
http//www.nishizeki.ecei.tohoku.ac.jp/nszk/saidur
/gdbook.html
109
ISAAC in Sendai, Tohoku
Probably 2007
110
Sendai
Kyoto
Tokyo
111
Thank You
112
Properties of a drawing of G(C)
x
x
y
z
z
y
orthogonal drawing of G(C)
minimum number of bends
the six open halflines are free
113
Main idea
114
Main idea
19
18
17
14
15
13
10
12
16
11
9
6
8
7
3
4
5
1
2
Add a group of vertices one by one.
4-canonical decomposition
115
Main idea
19
18
17
14
15
13
10
12
16
11
9
6
8
7
3
4
5
1
2
Add a group of vertices one by one.
4-canonical decomposition
116
Main idea
20
21
19
18
17
14
15
13
10
12
16
11
9
6
8
7
3
4
5
1
2
14-canonical decomposition
2 Find paths
O(n)NRN97
3 Decide x-coordinates
Time complexity O(n)
117
4-canonical decompositionNRN97
(a generalization of st-numbering)
G
118
4-canonical decompositionNRN97
(a generalization of st-numbering)
119
4-canonical decompositionNRN97
(a generalization of st-numbering)
120
4-canonical decompositionNRN97
(a generalization of st-numbering)
121
4-canonical decompositionNRN97
(a generalization of st-numbering)
face
U9
122
4-canonical decompositionNRN97
(a generalization of st-numbering)
123
4-canonical decompositionNRN97
(a generalization of st-numbering)
face
124
4-canonical decompositionNRN97
O(n)
(a generalization of st-numbering)
17
14
13
6
G
125
Orthogonal Drawings
i
f
g
h
e
g
h
9 bends
d
f
e
i
d
c
c
b
a
b
a
plane graph G
Input
Output
Each edge is drawn as an alternating sequence of
horizontal and vertical line segments.
g
f
Each vertex is drawn as a point.
h
Applications
e
i
d
Circuit schematics, Data-flow diagrams,
Entity-relationship diagrams T87, BK97.
a
c
b
4 bends
Objective
an orthogonal drawing with the minimum number of
bends.
To minimize the number of bends in an orthogonal
drawing.
126
Known Result
Garg and Tamassia GT96
time algorithm for finding an orthogonal
drawing of a plane graph of ? 4 with the
minimum number of bends.
Idea
reduction to a minimum cost flow problem
Rahman, Nakano and Nishizeki RNN99
A linear time algorithm to find an orthogonal
drawing of a 3-connected cubic plane graph with
the minimum number of bends.
Idea
reduction to a rectangular drawing problem.
Rahman and Nishizeki RN02
A linear time algorithm to find an orthogonal
drawing of a plane graph of ?3 with the
minimum number of bends.
Idea
reduction to a no-bend drawing problem.
127
Outline of the algorithm of RNN99
b
b
c
c
a
a
d
d
b
a
b
a
d
c
orthogonal drawings
rectangular drawing of
c
d
orthogonal drawing of G
Properties of a drawing of G(C)
x
x
y
z
z
y
orthogonal drawing of G(C)
minimum number of bends
the six open halflines are free
128
Rectangular Drawings
corner
Rectangular drawing of
G
Plane graph G of
Output
Input
Each vertex is drawn as a point.
Each edge is drawn as a horizontal or a vertical
line segment.
Each face is drawn as a rectangle.
Not every plane graph has a rectangular drawing.
Thomassen 84, Rahman, Nakano and Nishizeki 02
A necessary and sufficient condition.
Rahman, Nakano and Nishizeki 98, 02
A linear time algorithm
129
Rectangular Drawing Algorithm of RNN98
Input
Partition-pair
Splitting
130
Partition-pair
rectangular drawing of each subgraph
131
Rectangular Drawings
Modification of drawings patching
132
Plane multigraph
Input
Output

• Each vertex is drawn as a (possibly degenerated)
  rectangle on an integer grid.
• Each edge is drawn as a horizontal or a vertical
  line segment along grid line without bend.
• Each face is drawn as a rectangle.

Rahman, Nakano and Nishizeki RNN00
A necessary and sufficient condition.
A linear-time algorithm.
Reduce the problem to a rectangular drawing
problem.
133
Conclusions
We surveyed the recent algorithmic results on
various drawings of plane graphs.
Open Problems
Rectangular drawings of plane graphs of ? 4.
Efficient algorithm for bend-minimal orthogonal
drawings of plane graphs of ? 4
Parallel algorithms for drawing of plane graphs.
134
Application
VLSI floorplanning
Floor plan
Interconnection Graph
135
Insertion of four corners
Dual-like graph
136
Application
VLSI floorplanning
Floor plan
Interconnection Graph
137
st-numberingE79(2-connected graph)
(1) (1,n) is an edge of G
(2) Each vertex k, 2 lt k lt n-1, has at
least one neighbor and at least one upper neighbor


11
n12
10
9
8
7
5
4
6
3
2
1
138
Step 1 4-canonical ordering KH97
(4-connected graph)
(a generalization of st-numbering)
n18
n-117
(1) (1,2) and (n,n-1) are edges on the outer face
16
15
12
14
(2) Each vertex k, 3 lt k lt n-2, has at least
two lower neighbors and has at least two upper
neighbors
10
13
11
9


8
5
6
4
7
3
2
1
139
Step 2 Divide G into Gand G
n18
n-117
16
15
12
14
10
13
11
8
5
6
4
7
3
2
1
140
Step 3 Draw G in an isosceles right-angled
triangle
9
8
5
6
G
7
4
V n/2
3
2
1
slope lt 1

141
Step 3 Draw G in an isosceles right-angled
triangle
9
8
5
6
G
7
4
V n/2
3
2
1
1
2
???????? gt 1
???????? lt 1

x-??????
x-??
1???????????1 2?????1??2?????????????x-??
142
?????????
3
Initialization
2
1
1???????????1 2?????1??2?????????????x-??
143
The drawing of the path passing through all ks
neighbors is weakly convex
k
weakly convex
k
not weakly convex
144
The rightmost neighbor of k is higher than the
leftmost neighbor
Case 1k has exactly one highest neighbor
k
145
The rightmost neighbor of k is higher than the
leftmost neighbor
Case 2k has exactly two or more highest neighbor
k
146
Drawing of G
3
1
2
147
Drawing of G
shift
148
Drawing of G
149
Drawing of G
shift
150
Drawing of G
151
Drawing of G
shift
152
Drawing of G
153
Drawing of G
shift
154
Drawing of G
155
Drawing of G
shift
156
Drawing of G
157
Drawing of G
shift
158
Drawing of G
159
Drawing of G
G
Shift one time for each vertex
160
Step 5 Combine the drawing of G and G
W/2
1
W/2
W lt n/2 -1

161
Step 5 Combine the drawing of G and G
W/2
1
????gt1
W/2
W lt n/2 -1

???????? lt 1

162
Step 5 Combine the drawing of G and G
????gt1
???????? lt 1

163
Step 5 Combine the drawing of G and G
x-??
164
Step 5 Combine the drawing of G and G
W/2
1
W/2
W lt n/2 -1

165
Miura et al. 01
Input 4-connected plane graph G
Output a straight line grid drawing
Grid Size
H lt n/2

(W H lt
n-1) W lt
n/2 -1, H lt n/2



W lt n/2 -1
Area
W x H lt


The algorithm is best possible
H n/2
W n/2 -1
166
Objective
i
h
g
9 bends
g
f
h
f
e
i
e
d
d
a
c
b
c
4 bends
b
a
an orthogonal drawing with the minimum number of
bends.
To minimize the number of bends in an orthogonal
drawing.