Tohoku University

1
Tohoku University
2
Tohoku University
Sendai
360km
Tokyo
3
Inner Rectangular Drawings of Plane
Graphs?Application of Graph Drawing to VLSI
Layout-
Takao Nishizeki
Tohoku University
4
Inner Rectangular Drawing
an inner rectangular drawing of G
a plane graph G
1each vertex is drawn as a point
1each vertex is drawn as a point
2each edge is drawn as a horizontal or vertical
line segment
3all inner faces are drawn as rectangles
5
Inner Rectangular Drawing
an inner rectangular drawing of G
a plane graph G
1each vertex is drawn as a point
2each edge is drawn as a horizontal or vertical
line segment
2each edge is drawn as a horizontal or vertical
line segment
3all inner faces are drawn as rectangles
6
Inner Rectangular Drawing
rectilinear polygon
an inner rectangular drawing of G
a plane graph G
1each vertex is drawn as a point
2each edge is drawn as a horizontal or vertical
line segment
3all inner faces are drawn as rectangles
3all inner faces are drawn as rectangles
7
Application
VLSI floor planning
Ga
G
Vertex module edge adjacency among modules
8
Application
VLSI floor planning
The outer boundary of a VLSI chip is often an
axis-parallel polygon
2
1
1
3
4
5
6
7
9
8
Ga
G
Vertex module edge adjacency among modules
Inner rectangular drawing
9
Application
VLSI floor planning
The outer boundary of a VLSI chip is often an
axis-parallel polygon
2
2
1
2
3
4
5
6
7
9
8
Ga
G
Vertex module edge adjacency among modules
Inner rectangular drawing
10
Application
VLSI floor planning
The outer boundary of a VLSI chip is often an
axis-parallel polygon
2
2
1
1
3
4
5
6
7
9
8
Ga
G
Vertex module edge adjacency among modules
Inner rectangular drawing
11
Application
VLSI floor planning
dual-like graph
2
1
3
4
5
6
7
8
9
GGa
Ga
G
Vertex module edge adjacency among modules
12
Application
VLSI floor planning
The outer boundary of a VLSI chip is often an
axis-parallel polygon
dual-like graph
2
1
3
4
5
6
7
8
9
GGa
Ga
Vertex module edge adjacency among modules
13
Application
VLSI floor planning
The outer boundary of a VLSI chip is often an
axis-parallel polygon
dual-like graph
2
1
3
4
5
6
7
8
9
Ga
Vertex module edge adjacency among modules
14
Known Result
a necessary and sufficient condition for the
existence of a rectangular drawing of G with
T84,RNN98 and a linear algorithm for
RNN98,BS88,KH97
a plane graph G
15
Open Problem
a necessary and sufficient condition for the
existence of an inner rectangular drawing of G
(with )?
efficient algorithm to find an inner rectangular
drawing of G (with )?
?
a plane graph G
16
Our Results
1 a necessary and sufficient condition for the
existence of an inner rectangular drawing of
G.
17
Our Results
2 O(n1.5/log n) algorithm to find an inner
rectangular drawing of G if a sketch of
the outer face is given.
a plane graph G
a sketch of the outer face
18
Our Results
2 O(n1.5/log n) algorithm to find an inner
rectangular drawing of G if a sketch of
the outer face is given.
a plane graph G
19
Our Results
3 a polynomial time algorithm to find an inner
rectangular drawing of G in a general case,
where a sketch is not always given.
a plane graph G
20
1A necessary and sufficient condition for the
existence of an inner rectangular drawing of G.
2 O(n1.5 /log n) time algorithm to find an inner
rectangular drawing of G if a sketch of the outer
face is given.
3 a polynomial time algorithm to find an inner
rectangular drawing of G in a general case, where
a sketch is not always given.
21
Definition of Labeling
a plane graph G
Consider a labeling which assigns label 1,2 or 3
to every angle of G
22
Definition of Labeling
1 2 3
a plane graph G
Consider a labeling which assigns label 1,2 or 3
to every angle of G
23
Regular labeling
A regular labeling satisfies the following three
conditions (a)-(c)
(a) the labels of all the angles of each vertex v
total to 4
2
3
2
3
1
1
1
1
1
1
2
1
2
1
1
1
3
2
1
2
1
1
2
1
1
2
1
1
1
1
1
2
1
1
2
1
1
3
1
2
1
1
3
1
1
1
3
2
a plane graph G
an inner rectangular drawing of G
24
Regular labeling
(b) the labels of any inner angles is 1 or 2, and
any inner face has exactly four angles of
label 1
a plane graph G
an inner rectangular drawing of G
25
Regular labeling
(c) ncv - ncc 4. ncv the number of outer
angles having label 3 ncc the number of
outer angles having label 1
rectilinear polygon
convex corners
ncv6
a plane graph G
an inner rectangular drawing of G
26
Regular labeling
(c) ncv - ncc 4. ncv the number of outer
angles having label 3 ncc the number of
outer angles having label 1
convex corners
ncv6
ncc2
concave corners
a plane graph G
an inner rectangular drawing of G
27
Regular labeling
(c) ncv - ncc 4. ncv the number of outer
angles having label 3 ncc the number of
outer angles having label 1
ncv 4 ncc 0
ncv 5 ncc 1
ncv 6 ncc 2
28
A necessary and sufficient condition for the
existence of an inner rectangular drawing of
G
A plane graph G has an inner
rectangular drawing
29
A necessary and sufficient condition for the
existence of an inner rectangular drawing of
G
G has a regular labeling
30
A necessary and sufficient condition for the
existence of an inner rectangular drawing of
G
A plane graph G has an inner
rectangular drawing
G has a regular labeling
31
1A necessary and sufficient condition for the
existence of an inner rectangular drawing of G.
2 O(n1.5 /log n) time algorithm to find an inner
rectangular drawing of G if a sketch of the outer
face is given.
3 a polynomial time algorithm to find an inner
rectangular drawing of G in a general case, where
a sketch is not always given.
32
1A necessary and sufficient condition for the
existence of an inner rectangular drawing of G.
2 O(n1.5 /log n) time algorithm to find an inner
rectangular drawing of G if a sketch of the outer
face is given.
3 a polynomial time algorithm to find an inner
rectangular drawing of G in a general case, where
a sketch is not always given.
33
Inner rectangular drawing with sketched outer face
a plane graph G
Suppose that a sketch of the outer face of G is
prescribed, that is, all the outer angles of G
are labeled with 1, 2 or 3
34
Inner rectangular drawing with sketched outer face
a plane graph G
Suppose that a sketch of the outer face of G is
prescribed, that is, all the outer angles of G
are labeled with 1, 2 or 3
35
Inner rectangular drawing with sketched outer face
a plane graph G
Suppose that a sketch of the outer face of G is
prescribed, that is, all the outer angles of G
are labeled with 1, 2 or 3
36
Inner rectangular drawing with sketched outer face
a plane graph G
Suppose that a sketch of the outer face of G is
prescribed, that is, all the outer angles of G
are labeled with 1, 2 or 3
37
Inner rectangular drawing with sketched outer face
3
1
2
3
2
2
2
3
3
2
2
3
2
2
1
3
a plane graph G
Suppose that a sketch of the outer face of G is
prescribed, that is, all the outer angles of G
are labeled with 1, 2 or 3
Find an inner rectangular drawing with a
prescribed sketch of the outer face
38
Inner rectangular drawing with sketched outer face
3
1
2
3
2
2
2
3
3
2
2
3
2
2
1
3
a plane graph G
Suppose that a sketch of the outer face of G is
prescribed, that is, all the outer angles of G
are labeled with 1, 2 or 3
Find an inner rectangular drawing with a
prescribed sketch of the outer face
39
A plane graph G has an inner
rectangular drawing
G has a regular labeling
G has a regular labeling
Gd has a perfect matching
a plane graph G
40
Construct a decision graph Gd
Labels of some of the inner angles of G can be
immediately determined
a plane graph G
41
Construct a decision graph Gd
degree 2
a plane graph G
42
Construct a decision graph Gd
degree 2
2
3
3
3
1
2
2
2
3
1
3
3
2
a plane graph G
43
Construct a decision graph Gd
degree 4
a plane graph G
44
Construct a decision graph Gd
degree 4
a plane graph G
45
Construct a decision graph Gd
degree 4
a plane graph G
46
Construct a decision graph Gd
degree 4
a plane graph G
47
Construct a decision graph Gd
outer vertex of degree 2
2
2
1
1
a plane graph G
48
Construct a decision graph Gd
outer vertex of degree 2
2
2
1
1
a plane graph G
49
Construct a decision graph Gd
outer vertex of degree 2
2
2
1
1
a plane graph G
50
Construct a decision graph Gd
outer vertex of degree 3
2
2
a plane graph G
51
Construct a decision graph Gd
outer vertex of degree 3
2
2
1
1
a plane graph G
52
Construct a decision graph Gd
outer vertex of degree 3
1
2
2
1
1
1
1
1
1
1
a plane graph G
53
Construct a decision graph Gd
outer vertex of degree 3
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
a plane graph G
54
Construct a decision graph Gd
outer vertex of degree 3
1
a plane graph G
55
Construct a decision graph Gd
outer vertex of degree 3
a plane graph G
56
Construct a decision graph Gd
outer vertex of degree 3
a plane graph G
57
Construct a decision graph Gd
outer vertex of degree 3
a plane graph G
58
Construct a decision graph Gd
outer vertex of degree 3
a plane graph G
59
Construct a decision graph Gd
outer vertex of degree 3
x 1 or 2
a plane graph G
60
Construct a decision graph Gd
degree 3
x 1 or 2
a plane graph G
61
Construct a decision graph Gd
degree 3
x 1 or 2
a plane graph G
62
Construct a decision graph Gd
degree 3
x 1 or 2
a plane graph G
63
Construct a decision graph Gd
degree 3
x 1 or 2
a plane graph G
64
Construct a decision graph Gd
degree 3
2
1
1
1
1
1
1
1
1
x 1 or 2
a plane graph G
65
Construct a decision graph Gd
degree 3
x 1 or 2
a plane graph G
66
Construct a decision graph Gd
degree 3
2
1
1
1
1
1
1
1
1
x 1 or 2
a plane graph G
67
Construct a decision graph Gd
degree 3
x 1 or 2
a plane graph G
68
Construct a decision graph Gd
degree 3
x 1 or 2
a plane graph G
69
Construct a decision graph Gd
1
2
1
1
1
1
1
1
1
1
1
1
1
a plane graph G
70
Construct a decision graph Gd
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
71
Construct a decision graph Gd
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
72
Construct a decision graph Gd
2 of xs must be 1s.
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
73
Construct a decision graph Gd
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
74
Construct a decision graph Gd
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
75
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
76
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
77
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
78
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
79
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
80
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
81
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
82
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
83
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
84
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
85
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
86
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
2 of xs are 1s.
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
87
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
1
2
1
1
1
1
1
1
1
1
1
1
1
a decision graph Gd of G
a plane graph G
88
A necessary and sufficient condition for the
existence of a regular labeling
G has a regular labeling
Gd has a perfect matching
a decision graph Gd of G
89
A necessary and sufficient condition for the
existence of a regular labeling
A necessary and sufficient condition for the
existence of an inner rectangular drawing
G has an inner rectangular drawing with sketched
outer face
G has a regular labeling
Gd has a perfect matching
a decision graph Gd of G
90
Running time
nd O(n) md O(n)
A perfect matching of Gd can be found in time O(
) HK73,MV80 or in time O(
) FM91,Hoc04,HC04
a perfect matching of Gd
A perfect matching of Gd can be found in time
O(n1.5/log n)
91
Running time
nd O(n) md O(n)
A perfect matching of Gd can be found in time O(
) HK73,MV80 or in time O(
) FM91,Hoc04,HC04
92
Running time
nd O(n) md O(n)
A perfect matching of Gd can be found in time O(
) HK73,MV80 or in time O(
) FM91,Hoc04,HC04
An inner rectangular drawing of G can be found in
time O(n1.5/log n)
93
1A necessary and sufficient condition for the
existence of an inner rectangular drawing of G.
2 O(n1.5 /log n) time algorithm to find an inner
rectangular drawing of G if a sketch of the outer
face is given.
3 a polynomial time algorithm to find an inner
rectangular drawing of G in a general case, where
a sketch is not always given.
94
Case 1 the numbers of convex and concave outer
vertices are given.
ncv 6, ncc 2
95
Case 1 the numbers of convex and concave outer
vertices are given.
ncv 6, ncc 2
a plane graph G
96
Case 1 the numbers of convex and concave outer
vertices are given.
ncc - no4
ncv
ncv 6, ncc 2
a plane graph G
a decision graph Gd of G
97
Case 1 the numbers of convex and concave outer
vertices are given.
Running time
nd O(n) md O(N), Nn ncvno (no the
number of outer vertices). An inner rectangular
drawing of G can be found in time O(
).
98
Case 2 neither the outer sketch nor the numbers
of corners are given.
ncv , ncc are arbitrary
99
Case 2 neither the outer sketch nor the numbers
of corners are given.
a plane graph G
100
Case 2 neither the outer sketch nor the numbers
of corners are given.
no2 - no4 -4
a plane graph G
a decision graph Gd of G
101
Case 2 neither the outer sketch nor the numbers
of corners are given.
Running time
nd O(n) md O(N), N n (no2 no4 4)no
(no the number of outer vertices, no2 and no4
the numbers of outer vertices of degrees 2 and 4
) An inner rectangular drawing of G can be found
in time O( ).
102
Conclusion
(2) An inner rectangular drawing can be found
in time
O(n1.5 / log n) if the outer face is sketched.
O( ) if (ncv, ncc) is
prescribed.
Nn ncvno no the number of outer
vertices
O( ) for a general case.
N n (no2 no4 4)no no2 and no4 the
numbers of outer vertices of degrees 2 and 4

(3) Linear algorithm ?
103
(No Transcript)
104
Network Flow
2
3
1
2
1
3
3
1
1
2
1
1
3
2
1
2
2
2
1
2
1
1
3
2
Network N
G
105
Network Flow
2
3
1
1
0
2
1
1
1
1
3
0
3
2
1
2
1
1
1
1
0
2
2
4
1
1
0
2
1
1
3
2
1
2
1
1
1
0
2
2
1
2
1
2
1
1
3
2
Network N
G
106
Network Flow
2
3
1
1
0
2
1
1
2
1
1
1
3
0
3
2
1
1
2
1
1
2
1
1
2
1
1
0
2
2
1
4
1
1
0
2
1
1
3
2
1
1
2
1
2
1
1
1
1
2
1
0
2
2
1
2
1
2
1
1
3
2
Network N
G
107
Network Flow
2
3
1
2
1
2
1
3
3
1
1
1
2
1
2
1
2
1
1
1
3
1
2
1
2
1
1
1
2
2
2
2
1
2
1
1
3
2
G
108
(No Transcript)
109
Case 1 the numbers of convex and concave outer
vertices are given.
ncv 6, ncc 2
Case 2 general case
110
Inner rectangular drawing with prescribed numbers
ncv and ncc
ncv 6, ncc 2
a plane graph G
111
Inner rectangular drawing with prescribed numbers
ncv and ncc
ncv 6, ncc 2
a plane graph G
a decision graph Gd of G
112
Inner rectangular drawing with prescribed numbers
ncv and ncc
ncc - no4
ncv
ncv 6, ncc 2
y
y
x
x
y
1
1
x
1
1
x
x
x
x
x
x
y
2
2
x
x
x
x
x
x
x
x
x
x
1
x
1
x
x
x
1
1
y
x
x
x
x
x
y
x
x
y
x
a plane graph G
a decision graph Gd of G
113
Inner rectangular drawing with prescribed numbers
ncv and ncc
ncc - no4
ncv
ncv 6, ncc 2
y
y
x
x
y
1
1
x
1
1
x
x
x
x
x
x
y
2
2
x
x
x
x
x
x
x
x
x
x
1
x
1
x
x
x
1
1
y
x
x
x
x
x
y
x
x
y
x
a plane graph G
a decision graph Gd of G
114
Inner rectangular drawing with prescribed numbers
ncv and ncc
ncv 6, ncc 2
y
y
3
3
x
x
y
3
1
1
x
1
1
x
x
x
x
x
x
y
2
3
2
x
x
x
x
x
2
x
x
x
1
x
x
1
x
1
x
x
x
1
1
y
3
2
x
x
x
x
x
y
x
x
y
3
x
a plane graph G
115
Running time
Gd has an O(n) number of vertices and O(N) (Nn
ncvno no the number of outer vertices) number
of edges. An inner rectangular drawing D of G can
be found in time O( ).
116
Case 1 the numbers of convex and concave outer
vertices are given.
ncv 6, ncc 2
Case 2 in general case
117
Inner rectangular drawing
no2 - no4 -4
y
y
x
x
y
1
1
x
1
1
x
x
x
x
x
x
y
2
2
x
x
x
x
x
x
x
x
x
x
1
x
1
x
x
x
1
1
y
x
x
x
x
x
y
x
x
y
x
a plane graph G
a decision graph Gd of G
118
Inner rectangular drawing
no2 - no4 -4
y
y
x
x
y
1
1
x
1
1
x
x
x
x
x
x
y
2
2
x
x
x
x
x
x
x
x
x
x
1
x
1
x
x
x
1
1
y
x
x
x
x
x
y
x
x
y
x
a plane graph G
a decision graph Gd of G
119
Inner rectangular drawing
120
Running time
Gd has an O(n) number of vertices and O(N)
(Nn (no2 - no4 4)no no the number of
outer vertices no2 and no4 the numbers of
outer vertices of degrees 2 and 4 )
number of edges. An inner rectangular drawing D
of G can be found in time O(
).
121
(No Transcript)
122
Related result
If a sketch of several faces of G including the
outer face is prescribed, then one can examine
whether G has a drawing such that each of the
other face is a rectangle.
123
Related result
If faces F0,F1, Fi of G are vertex-disjoint and
the numbers of convex and concave vertices are
prescribed, then one can examine whether G has a
drawing such that each of F0,F1, Fi has
prescribed numbers of convex and concave vertices
and each of the other faces is a rectangle.
124
Regular labeling
We call f a regular labeling of G if f
satisfies the following three conditions (a)-(c)
(a) the labels of any vertex in G total to 4
(b) the labels of any inner angles is 1 or 2, and
any inner face has exactly four angles of
label 1
(c) ncv - ncc 4. ncv the number of outer
angles having label 3 ncc the number of
outer angles having label 1
125
A necessary and sufficient condition for the
existence of an inner rectangular drawing of
G
A plane graph G has an inner rectangular drawing
if and only if G has a regular labeling
a plane graph G
126
Construct a decision graph Gd
Some of the inner angles of G can be immediately
determined
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a plane graph G
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Construct a decision graph Gd
Some of the inner angles of G can be immediately
determined