Download - Drawing Plane Graphs
![Page 1: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/1.jpg)
Drawing Plane Graphs
Takao Nishizeki
Tohoku University
![Page 2: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/2.jpg)
US PresidentCalifornia Governor
![Page 3: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/3.jpg)
US PresidentCalifornia Governor
What is the common feature?
![Page 4: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/4.jpg)
Graphs and Graph Drawings
A diagram of a computer network
ATM-SW
ATM-SW
ATM-SW
ATM-SW
ATM-SW
ATM-HUBATM-RT
TPDDI
ATM-RT
ATM-HUB
ATM-HUB
ATM-RT
TPDDI
ATM-RT
ATM-HUB
STATIONSTATION
STATIONSTATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATIONTPDDI
STATIONATM-HUB
STATION
STATION
![Page 5: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/5.jpg)
Objectives of Graph Drawings
To obtain a nice representation of a graph so that the structure of the graph is easily understandable.
structure of the graph is difficult to understand
structure of the graph is easy to understand
Nice drawing
Symmetric Eades, Hong
![Page 6: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/6.jpg)
Objectives of Graph Drawings
Nice drawing
Ancient beauty
Modernbeauty
![Page 7: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/7.jpg)
The drawing should satisfy some criterion arising from the application point of view.
1 2 3
4
57
8
not suitable for single layered PCB
suitable for single layered PCB
1 2 3
4
6
8
57
Objectives of Graph Drawings
Diagram of an electronic circuit
Wire crossings
![Page 8: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/8.jpg)
Drawings of Plane Graphs
Convex drawing
Straight line drawing
![Page 9: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/9.jpg)
Drawings of Plane Graphs
Rectangular drawing
Orthogonal drawing
Box-rectangular drawing
![Page 10: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/10.jpg)
Book
Planar Graph Drawing
by
Takao NishizekiMd. Saidur Rahman
http://www.nishizeki.ecei.tohoku.ac.jp/nszk/saidur/gdbook.html
![Page 11: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/11.jpg)
Straight Line Drawing
Plane graph
![Page 12: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/12.jpg)
Straight Line Drawing
Plane graphStraight line drawing
![Page 13: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/13.jpg)
Straight Line Drawing
Each vertex is drawn as a point.
Plane graphStraight line drawing
![Page 14: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/14.jpg)
Straight Line Drawing
Each vertex is drawn as a point.
Each edge is drawn as a single straight line segment.
Plane graphStraight line drawing
![Page 15: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/15.jpg)
Straight Line Drawing
Each vertex is drawn as a point.
Each edge is drawn as a single straight line segment.
Plane graphStraight line drawing
Every plane graph has a straight line drawing.
Wagner ’36 Fary ’48Polynomial-time algorithm
![Page 16: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/16.jpg)
Straight Line Drawing
Plane graphStraight line drawing
W
H
W
H
![Page 17: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/17.jpg)
Straight Line Grid Drawing
In a straight line grid drawing each vertex is drawn on a grid point.
Plane graph
Straight line grid drawing.
![Page 18: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/18.jpg)
Wagner ’36 Fary ’48
Plane graph
Straight line grid drawing.
Grid-size is not polynomial of the number of vertices n
![Page 19: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/19.jpg)
Plane graph
Straight line grid drawing.
de Fraysseix et al. ’9022nHW
nW 2
nH
Straight Line Grid Drawing
![Page 20: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/20.jpg)
Plane graph
Straight line grid drawing.
de Fraysseix et al. ’9022nHW
nW 2
nH
Chrobak and Payne ’95
Linear algorithm
![Page 21: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/21.jpg)
n-2
n-2
Schnyder ’90
W
H
2nHW Upper bound
![Page 22: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/22.jpg)
What is the minimum size of a grid required for a straight line drawing?
![Page 23: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/23.jpg)
2
3
2
nHW
Lower Bound
nH3
2
nW3
2
![Page 24: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/24.jpg)
A restricted class of plane graphs may have more compact grid drawing.
Triangulated plane graph
3-connected graph
![Page 25: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/25.jpg)
4-connected ?
not 4-connected disconnected
![Page 26: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/26.jpg)
How much area is required for 4-connected plane graphs?
![Page 27: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/27.jpg)
W < n/2
H < n/2
=
=
Straight line grid drawing
Miura et al. ’01
Input: 4-connected plane graph G
Output: a straight line grid drawing Grid Size :
Area:2
,n
HW
4
2nHW
![Page 28: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/28.jpg)
n-2
n-2
Area < 1/4=
Area=n2 Area<n /42
W < n/2
H < n/2
=
=
.. =
plane graph G 4-connected plane graph G
Schnyder ’90 Miura et al. ’01
![Page 29: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/29.jpg)
The algorithm of Miura et al. isbest possible
2
nH
2
nW
4
2nHW
![Page 30: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/30.jpg)
|slope|>1|slope|>1|slope|>1
Main idea
G
Step2: Divide G into two halves G’ and G”
n/2=9
G”
G’1 2
n-1=17 n=18
347
8 56
1410
15 16
1311
12
W < n/2 -1=
W/2
W/2
1
G’
G”
Step3 and 4 : Draw G’ andG” in isosceles right-angledtriangles
G
n/2
n/2 -1
Step5: Combine the drawings of G’ and G”
Triangulate all inner faces
1 2
n-1=17 n=18
347
8 56
14
910
15 16
1311
12
Step1: find a 4-canonical ordering
|slope|<1=
![Page 31: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/31.jpg)
Straight line drawing
Convex drawing
Draw a graph G on the plane “nicely”
A convex drawing is a straight line drawing where each face is drawn as a convex polygon.
![Page 32: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/32.jpg)
Convex drawing
Convex Drawing
Tutte 1963Every 3-connected planar graph has a convex drawing.A necessary and sufficient condition for a plane graph to have a convex drawing. Thomassen ’80
![Page 33: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/33.jpg)
Convex drawing
Convex Drawing
Tutte 1963Every 3-connected planar graph has a convex drawingA necessary and sufficient condition for a plane graph to have a convex drawing. Thomassen ’80
Chiba et al. ’84
O(n) time algorithm
![Page 34: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/34.jpg)
n-2
n-2
Convex Grid Drawing
Chrobak and Kant ’97
2nHW
Input: 3-connected graph
Output: convex grid drawing
Area
Grid Size
![Page 35: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/35.jpg)
Input : 4-connected plane graph
W
H
Miura et al. 2000
Output: Convex grid drawing
1 nHW
4
2nHW
Half-perimeter
Area
Convex Grid Drawing
Grid Size
![Page 36: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/36.jpg)
Area < 1/4=
3-connected graph 4-connected graph
Area
n-2
n-2
W
H
Area
Chrobak and Kant ’97 Miura et al. 2000
2n4
2n
![Page 37: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/37.jpg)
W
H
The algorithm of Miura et al. isbest possible
4
2nHW
![Page 38: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/38.jpg)
1: 4-canonical decomposition1 2
34 5
67 8
9
1011 12
131415
16
1718 19
20 21
13
1415
16 1718 19
20 21
11 12
9 107 86
3 4 54: Decide y-coordinates
1 2 Time complexity: O(n)
916
1 2
34 5
67
8
14
21
12
13
20
10
19
11
1715
18
O(n)[NRN97]2: Find paths
3: Decide x-coordinates
Main idea
![Page 39: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/39.jpg)
EA
B
C
F
G
D
VLSI FloorplanningVLSI Floorplanning
Interconnection graph
![Page 40: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/40.jpg)
EA
B
C
F
G
D
AB
EC
F
G
D
VLSI FloorplanningVLSI Floorplanning
Interconnection graph VLSI floorplan
![Page 41: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/41.jpg)
EA
B
C
F
G
D
AB
EC
F
G
D
VLSI FloorplanningVLSI Floorplanning
Interconnection graph VLSI floorplan
![Page 42: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/42.jpg)
EA
B
C
F
G
D
AB
EC
F
G
D
VLSI FloorplanningVLSI Floorplanning
Interconnection graph VLSI floorplan
![Page 43: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/43.jpg)
EA
B
C
F
G
D
AB
EC
F
G
D
VLSI FloorplanningVLSI Floorplanning
AB
E
C
F
G
D
Interconnection graph VLSI floorplan
Dual-like graph
![Page 44: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/44.jpg)
EA
B
C
F
G
D
AB
EC
F
G
D
VLSI FloorplanningVLSI Floorplanning
AB
E
C
F
G
D
AB
E
C
F
G
D
Interconnection graph VLSI floorplan
Dual-like graph Add four corners
![Page 45: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/45.jpg)
EA
B
C
F
G
D
AB
EC
F
G
D
VLSI FloorplanningVLSI Floorplanning
AB
E
C
F
G
D
AB
E
C
F
G
D
Interconnection graph VLSI floorplan
Dual-like graph Add four corners
Rectangular drawing
![Page 46: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/46.jpg)
Rectangular Drawings
Plane graph G of 3
Input
![Page 47: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/47.jpg)
Rectangular Drawings
Rectangular drawing of GPlane graph G of 3
Input Output
corner
![Page 48: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/48.jpg)
Rectangular Drawings
Rectangular drawing of GPlane graph G of 3
Each vertex is drawn as a point.Input Output
corner
![Page 49: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/49.jpg)
Rectangular Drawings
Rectangular drawing of GPlane graph G of 3
Each edge is drawn as a horizontal or a vertical line segment.
Each vertex is drawn as a point.Input Output
corner
![Page 50: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/50.jpg)
Rectangular Drawings
Rectangular drawing of GPlane graph G of 3
Each edge is drawn as a horizontal or a vertical line segment.
Each face is drawn as a rectangle.
Each vertex is drawn as a point.Input Output
corner
![Page 51: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/51.jpg)
Not every plane graph has a rectangular drawing.
![Page 52: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/52.jpg)
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
![Page 53: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/53.jpg)
G
G has a rectangular drawing if and only if
2-legged cycles 3-legged cycles
A Necessary and Sufficient Condition by Thomassen ’84A Necessary and Sufficient Condition by Thomassen ’84plane graphfour vertices of degree 2 are designated as cornersfour vertices of degree 2 are designated as corners
every 2-legged cycle in every 2-legged cycle in GG contains at least two contains at least two designated vertices; anddesignated vertices; and
every 3-legged cycle in every 3-legged cycle in GG contains at least one contains at least one designated vertex.designated vertex.
![Page 54: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/54.jpg)
G
G has a rectangular drawing if and only if
2-legged cycles
A Necessary and Sufficient Condition by Thomassen ’84A Necessary and Sufficient Condition by Thomassen ’84plane graphfour vertices of degree 2 are designated as cornersfour vertices of degree 2 are designated as corners
every 2-legged cycle in every 2-legged cycle in GG contains at least two contains at least two designated vertices; anddesignated vertices; and
every 3-legged cycle in every 3-legged cycle in GG contains at least one contains at least one designated vertex.designated vertex.
![Page 55: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/55.jpg)
G
G has a rectangular drawing if and only if
2-legged cycles 3-legged cycles
A Necessary and Sufficient Condition by Thomassen ’84A Necessary and Sufficient Condition by Thomassen ’84plane graphfour vertices of degree 2 are designated as cornersfour vertices of degree 2 are designated as corners
every 2-legged cycle in every 2-legged cycle in GG contains at least two contains at least two designated vertices; anddesignated vertices; and
every 3-legged cycle in every 3-legged cycle in GG contains at least one contains at least one designated vertex.designated vertex.
![Page 56: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/56.jpg)
G
G has a rectangular drawing if and only if
2-legged cycles 3-legged cycles
A Necessary and Sufficient Condition by Thomassen ’84A Necessary and Sufficient Condition by Thomassen ’84plane graphfour vertices of degree 2 are designated as cornersfour vertices of degree 2 are designated as corners
every 2-legged cycle in every 2-legged cycle in GG contains at least two contains at least two designated vertices; anddesignated vertices; and
every 3-legged cycle in every 3-legged cycle in GG contains at least one contains at least one designated vertex.designated vertex. Bad cycles
![Page 57: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/57.jpg)
Outline of the Algorithm of Rahman et al.
![Page 58: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/58.jpg)
Outline of the Algorithm of Rahman et al.
![Page 59: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/59.jpg)
Outline of the Algorithm of Rahman et al.
![Page 60: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/60.jpg)
Outline of the Algorithm of Rahman et al.
![Page 61: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/61.jpg)
Outline of the Algorithm of Rahman et al.
![Page 62: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/62.jpg)
Outline of the Algorithm of Rahman et al.
![Page 63: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/63.jpg)
![Page 64: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/64.jpg)
Bad cycle
![Page 65: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/65.jpg)
Bad cycle
Bad cycle
![Page 66: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/66.jpg)
Partition-pair
![Page 67: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/67.jpg)
Partition-pair Splitting
![Page 68: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/68.jpg)
Partition-pair
![Page 69: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/69.jpg)
Partition-pair
![Page 70: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/70.jpg)
Partition-pair
Rectangular drawing
![Page 71: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/71.jpg)
Partition-pair
Rectangular drawing
Time complexity
O(n)
![Page 72: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/72.jpg)
Miura, Haga, N. ’03, Working paper
Rectangular drawing of plane graph G with Δ ≤ 4
perfect matching in Gd
![Page 73: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/73.jpg)
G
![Page 74: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/74.jpg)
Gd
G
![Page 75: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/75.jpg)
Gd perfect matching
G
![Page 76: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/76.jpg)
G )( 5.1nO
![Page 77: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/77.jpg)
EA
B
C
F
G
D
AB
EC
F
G
D
VLSI FloorplanningVLSI Floorplanning
Interconnection graph VLSI floorplan
Rectangular drawing
![Page 78: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/78.jpg)
EA
B
C
F
G
D
AB
EC
F
G
D
VLSI FloorplanningVLSI Floorplanning
Interconnection graph VLSI floorplan
Rectangular drawing
Unwanted adjacency
Not desirable for MCM floorplanning andfor some architectural floorplanning.
![Page 79: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/79.jpg)
A
B
E C
F
G
D
MCM FloorplanningMCM Floorplanning SherwaniSherwani
Architectural FloorplanningArchitectural Floorplanning Munemoto, Katoh, ImamuraMunemoto, Katoh, Imamura
EA
B
C
F
G
D
Interconnection graph
![Page 80: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/80.jpg)
A
B
E C
F
G
D
MCM FloorplanningMCM FloorplanningArchitectural FloorplanningArchitectural Floorplanning
EA
B
C
F
G
D
Interconnection graph
![Page 81: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/81.jpg)
MCM FloorplanningMCM FloorplanningArchitectural FloorplanningArchitectural Floorplanning
A
E
B
FG
CD
EA
B
C
F
G
D
Interconnection graph
Dual-like graph
A
B
E C
F
G
D
![Page 82: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/82.jpg)
MCM FloorplanningMCM FloorplanningArchitectural FloorplanningArchitectural Floorplanning
A
E
B
FG
CD
EA
B
C
F
G
D
Interconnection graph
Dual-like graph
A
B
E C
F
G
D
![Page 83: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/83.jpg)
MCM FloorplanningMCM FloorplanningArchitectural FloorplanningArchitectural Floorplanning
A
E
B
FG
CD
EA
B
C
F
G
D
A
E
B
FG
CD
Interconnection graph
Dual-like graph
A
B
E C
F
G
D
![Page 84: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/84.jpg)
MCM FloorplanningMCM FloorplanningArchitectural FloorplanningArchitectural Floorplanning
A
E
B
FG
CD
EA
B
C
F
G
D
A
E
B
FG
CD
Box-Rectangular drawing
Interconnection graph
Dual-like graph
A
B
E C
F
G
D
dead space
![Page 85: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/85.jpg)
Box-Rectangular Drawing
InputPlane multigraph G
ab
cd
![Page 86: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/86.jpg)
Box-Rectangular Drawing
Input OutputPlane multigraph G
ab
cd
a b
cdBox-rectangular drawing
![Page 87: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/87.jpg)
Box-Rectangular Drawing
Input
• Each vertex is drawn as a rectangle.
Plane multigraph G
ab
cd
a b
cd
Output
Box-rectangular drawing
![Page 88: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/88.jpg)
Box-Rectangular Drawing
Input
• Each vertex is drawn as a rectangle.• Each edge is drawn as a horizontal or a vertical line
segment.
Plane multigraph G
ab
cd
a b
cd
Output
Box-rectangular drawing
![Page 89: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/89.jpg)
Box-Rectangular Drawing
Input
• 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.
Plane multigraph G
ab
cd
a b
cd
Output
Box-rectangular drawing
![Page 90: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/90.jpg)
A necessary and sufficient condition for a plane multigraph to have a box-rectangular drawing.
A linear-time algorithm.
Rahman et al. 2000
2 mHW , where m is the number of edges in G.
W
H
a b
cd
![Page 91: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/91.jpg)
Algorithm of Rahman et al.
Main Idea: Reduction to a rectangular drawing problem
![Page 92: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/92.jpg)
G
Outline
![Page 93: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/93.jpg)
G
Outline
G
Replace each vertex of degree four or more by a cycle
![Page 94: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/94.jpg)
G
Outline
G linear time[RNN98a]
Rectangular drawing
![Page 95: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/95.jpg)
G
Outline
G linear time[RNN98a]
Box-rectangular drawing
![Page 96: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/96.jpg)
G
Outline
G linear time[RNN98a]
Box-rectangular drawing
Time complexity
O(n)
![Page 97: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/97.jpg)
Orthogonal Drawings
Input
plane graph Ga b
c
d
efg
h
i
![Page 98: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/98.jpg)
Orthogonal Drawings
Input Output plane graph G
a bc
d
efg
h
i
a bc
def
gh
i
![Page 99: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/99.jpg)
Orthogonal Drawings
Each edge is drawn as an alternating sequence of horizontal and vertical line segments.
Input Output plane graph G
bends
a bc
d
efg
h
i
a bc
def
gh
i
![Page 100: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/100.jpg)
Orthogonal Drawings
Each edge is drawn as an alternating sequence of horizontal and vertical line segments.Each vertex is drawn as a point.
Input Output plane graph G
bend
a bc
d
efg
h
i
a bc
def
gh
i
![Page 101: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/101.jpg)
Applications
Entity-relationship diagramsFlow diagrams
![Page 102: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/102.jpg)
Applications
Circuit schematics
Minimization of bends reduces the number of “vias” or “throughholes,” and hence reduces VLSI fabrication costs.
![Page 103: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/103.jpg)
plane graph Ga b
c
d
efg
h
i
a bc
def
gh
iObjectiveObjective
4 bends
de
a b c
fg
hi
minimum number of bends.
9 bend
To minimize the number of bends in an orthogonal drawing.
![Page 104: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/104.jpg)
Garg and Tamassia ’96
O n n( log )/ /7 4 1 2 time for plane graph of Δ ≤4
Idea reduction to a minimum cost flow problem
for 3-connected cubic plane graph Idea reduction to a rectangular drawing problem.
Rahman, Nakano and Nishizeki ’99
Bend-Minimum Orthogonal Drawing
Linear
Rahman and Nishizeki ’02
Linear for plane graph of Δ≤3
![Page 105: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/105.jpg)
Outline of the algorithm of [RNN99]
G G
1C
2Ca
bc
dG
)( 1CG
)( 2CG
a
b
c
d
orthogonal drawings
b
rectangular drawing of G
a
cd
orthogonal drawing of G
a b
cd
![Page 106: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/106.jpg)
Open Problems and Future Research Direction
More area-efficient straight-line or convex drawing algorithms.
Prescribed face areas
Prescribed length of some edgesPrescribed positions of some vertices
Practical Applications
Linear algorithm for rectangular drawings of plane graphs of Δ ≤4.
Drawing of plane graphs with constraints like
2
3
2
nHW for every planar graph?
Linear algorithm for bend-minimal orthogonal drawings of plane graphs of Δ ≤4.
![Page 107: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/107.jpg)
Book
Planar Graph Drawing
by
Takao NishizekiMd. Saidur Rahman
http://www.nishizeki.ecei.tohoku.ac.jp/nszk/saidur/gdbook.html
![Page 108: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/108.jpg)
Book
Planar Graph Drawing
by
Takao NishizekiMd. Saidur Rahman
http://www.nishizeki.ecei.tohoku.ac.jp/nszk/saidur/gdbook.html
![Page 109: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/109.jpg)
ISAAC in Sendai, Tohoku
Probably 2007
![Page 110: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/110.jpg)
SendaiKyoto
Tokyo
![Page 111: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/111.jpg)
![Page 112: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/112.jpg)
Properties of a drawing of G(C)
minimum number of bendsthe six open halflines are free
G(C)
orthogonal drawing of G(C)
x
yz
x
yz
![Page 113: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/113.jpg)
Main idea
4-canonical decomposition1 2
3
4 5
67 8
9
1011 12
131415
16
1718 19
20 21
![Page 114: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/114.jpg)
Main idea
4-canonical decomposition1 2
3
4 5
67 8
9
1011 12
131415
16
1718 19
20 21
13
1415
16 1718 19
20 21
11 12
9 107 86
3 4 5
1 21 2
3
4 5
67 8
9
10
20 21
Add a group of vertices one by one.
![Page 115: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/115.jpg)
Add a group of vertices one by one.
Main idea
4-canonical decomposition1 2
3
4 5
67 8
9
1011 12
131415
16
1718 19
20 21
13
1415
16 1718 19
20 21
11 12
9 107 86
3 4 5
1 2
![Page 116: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/116.jpg)
1:4-canonical decomposition1 2
34 5
67 8
9
1011 12
131415
16
1718 19
20 21
13
1415
16 1718 19
20 21
11 12
9 107 86
3 4 54: Decide y-coordinates
1 2 Time complexity: O(n)
916
1 2
34 5
67
8
14
21
12
13
20
10
19
11
1715
18
O(n)[NRN97]2: Find paths
3: Decide x-coordinates
Main idea
![Page 117: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/117.jpg)
U3
Gk
Gk -1
(a)
Gk
Gk -1
Gk
Gk -1
(b)
(c)
face
face
Uk
Uk
Uk
U12
U11
U10
U9
U8
U7
U6U5
U4
U2
4-canonical decomposition[NRN97]
U1
(a generalization of st-numbering)
G
![Page 118: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/118.jpg)
U3
4-canonical decomposition[NRN97](a generalization of st-numbering)
Gk -1
Gk
Gk
Gk -1
(a)
Gk
Gk -1
Gk
Gk -1
(b)
(c)
face
face
![Page 119: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/119.jpg)
Gk
Gk -1
(a)
Gk
Gk -1
Gk
Gk -1
(b)
(c)
face
face
U12
U11
U10
U9
U8
U7
U6U5
U4
U3
U2
4-canonical decomposition[NRN97]
U1
(a generalization of st-numbering)
![Page 120: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/120.jpg)
U9
4-canonical decomposition[NRN97](a generalization of st-numbering)
Gk -1
Gk
face
Gk
Gk -1
(a)
Gk
Gk -1
Gk
Gk -1
(b)
(c)
face
face
![Page 121: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/121.jpg)
Gk
Gk -1
(a)
Gk
Gk -1
Gk
Gk -1
(b)
(c)
face
face
U12
U11
U10
U9
U8
U7
U6U5
U4
U3
U2
4-canonical decomposition[NRN97]
U1
(a generalization of st-numbering)
![Page 122: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/122.jpg)
U5
4-canonical decomposition[NRN97](a generalization of st-numbering)
Gk -1
Gk
face
Gk
Gk -1
(a)
Gk
Gk -1
Gk
Gk -1
(b)
(c)
face
face
![Page 123: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/123.jpg)
U1
U12
U11
U10
U9
U8
U7
U6U5
U4
U3
U2
4-canonical decomposition[NRN97]
1 2
3
4 5
67
89 10
11
1213 1415
16
1718 19
20 n = 21
(a generalization of st-numbering)
Gk
Gk -1
(a)
Gk
Gk -1
Gk
Gk -1
(b)
(c)
face
face
Uk
Uk
Uk
G
O(n)
![Page 124: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/124.jpg)
Orthogonal Drawings
Each edge is drawn as an alternating sequence of horizontal and vertical line segments.
Each vertex is drawn as a point.
ApplicationsApplicationsCircuit schematics, Data-flow diagrams, Entity-relationship diagrams [T87, BK97].
ObjectiveObjectiveTo minimize the number of bends in an orthogonal drawing.
Input Output plane graph G
an orthogonal drawing with the minimum number of bends.
9 bends
4 bends
a bc
d
efg
h
i
a bc
def
gh
i
de
a b c
fghi
![Page 125: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/125.jpg)
Known ResultKnown ResultGarg and Tamassia [GT96]O n n( log )/ /7 4 1 2
time algorithm for finding an orthogonal drawing of a plane graph of Δ ≤4 with the minimum number of bends.
Idea reduction to a minimum cost flow problem
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, Nakano and Nishizeki [RNN99]
A linear time algorithm to find an orthogonal drawing of a plane graph of Δ≤3 with the minimum number of bends.
Idea reduction to a no-bend drawing problem.
Rahman and Nishizeki [RN02]
![Page 126: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/126.jpg)
Outline of the algorithm of [RNN99]
G G G
)( 1CG
)( 2CG
1C
2C
rectangular drawing of
G orthogonal drawings
orthogonal drawing of G
Properties of a drawing of G(C)
minimum number of bendsthe six open halflines are free
G(C)
orthogonal drawing of G(C)
a
bc
d
a
bc
d
ab
cd
a b
cd
x
yz
x
yz
![Page 127: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/127.jpg)
Rectangular Drawings
Rectangular drawing of GPlane graph G of 3
Each edge is drawn as a horizontal or a vertical line segment.
Each face is drawn as a rectangle.
Each vertex is drawn as a point.Input Output
corner
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
![Page 128: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/128.jpg)
Rectangular Drawing Algorithm of [RNN98]
Splitting
Partition-pairInput
![Page 129: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/129.jpg)
rectangular drawing of each subgraph
Partition-pair
![Page 130: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/130.jpg)
Modification of drawingspatching
Rectangular Drawings
![Page 131: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/131.jpg)
Box-Rectangular Drawing
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.
Plane multigraph
Reduce the problem to a rectangular drawing problem.
![Page 132: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/132.jpg)
Conclusions
We surveyed the recent algorithmic results on various drawings of plane graphs.
Open Problems
Rectangular drawings of plane graphs of Δ ≤ 4.
Efficient algorithm for bend-minimal orthogonal drawings of plane graphs of Δ ≤ 4
Parallel algorithms for drawing of plane graphs.
![Page 133: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/133.jpg)
1
2
3
4
5
678
9
10
1112
1314
1516
17
12
3
4
5
6
7
89
1017
1112
13
14
1516
Application VLSI floorplanning
Interconnection Graph Floor plan
![Page 134: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/134.jpg)
1
23
4
5
6789
10
11 12
13
14
1516
17
1
23
4
5
6789
10
11 12
13
14
1516
17
Dual-like graph Insertion of four corners
![Page 135: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/135.jpg)
1
2
3
4
5
678
9
10
1112
1314
1516
17
12
3
4
5
6
7
89
1017
1112
13
14
1516
Application VLSI floorplanning
Interconnection Graph Floor plan
![Page 136: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/136.jpg)
st-numbering[E79](2-connected graph )
(1) (1,n) is an edge of G(2) Each vertex k, 2 < k < n-1, has at least one neighbor and at least one upper neighbor
1 2
n=12
3
4
78
5
6
10
11
9
= =
![Page 137: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/137.jpg)
1 2
3
47
8 56
1410
1516
1311
12
9
Step 1: 4-canonical ordering [KH97] (4-connected graph)
(1) (1,2) and (n,n-1) are edges on the outer face
(2) Each vertex k, 3 < k < n-2, has at least two lower neighbors and has at least two upper neighbors
n-1=17 n=18
= =
(a generalization of st-numbering)
![Page 138: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/138.jpg)
G’’
G’ |V’|= n/2
|V”|= n/2
Step 2: Divide G into G’and G”
1 2
n-1=17 n=18
3
47
8 56
1410
1516
1311
12
n/2 =9
![Page 139: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/139.jpg)
Step 3: Draw G’ in an isosceles right-angled triangle
G
1 2
347
8 56
9
G’ |V’|= n/2
|slope| < 1 =
1 2
|slope| > 1
1 2
W/2
W < n/2 -1=
![Page 140: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/140.jpg)
G
1 2
347
8 56
9
G’ |V’|= n/2
1 :外周上の辺の傾きは高々 12 :外周上で点 1 から 2 へ時計回りに進む道の描画は x- 単調
| 外周上の辺の傾き | < 1 =
x- 単調
1 2| 外周上の辺の傾き | > 1
x- 単調ではない
1 2
Step 3: Draw G’ in an isosceles right-angled triangle
![Page 141: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/141.jpg)
shift and put k
各点の座標の決め方
1 2
3
G3Initialization :
k
Gk
1 :外周上の辺の傾きは高々 12 :外周上で点 1 から 2 へ時計回りに進む道の描画は x- 単調
各点 k, 4 < k < n/2 = =
k
Gk -1
![Page 142: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/142.jpg)
The drawing of the path passing through all k’s neighbors is “weakly convex”
k
k
“weakly convex”
not “weakly convex”
![Page 143: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/143.jpg)
k
Gk -1
Case 1 : k has exactly one highest neighbor
shift and put k
Gk
k
The rightmost neighbor of k is higher than the leftmost neighbor
![Page 144: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/144.jpg)
shift and put k
k
Gk
k
Gk -1
The rightmost neighbor of k is higher than the leftmost neighbor
Case 2 : k has exactly two or more highest neighbor
![Page 145: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/145.jpg)
Drawing of G’
1 2
3
G3
![Page 146: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/146.jpg)
shift
Drawing of G’
![Page 147: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/147.jpg)
+4
Drawing of G’
![Page 148: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/148.jpg)
shift
Drawing of G’
![Page 149: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/149.jpg)
+5
Drawing of G’
![Page 150: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/150.jpg)
shift
Drawing of G’
![Page 151: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/151.jpg)
+6
Drawing of G’
![Page 152: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/152.jpg)
shift
Drawing of G’
![Page 153: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/153.jpg)
+7
Drawing of G’
![Page 154: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/154.jpg)
shift
Drawing of G’
![Page 155: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/155.jpg)
+8
Drawing of G’
![Page 156: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/156.jpg)
shift
Drawing of G’
![Page 157: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/157.jpg)
+9
Drawing of G’
![Page 158: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/158.jpg)
G’
H’ <W’/2=( n/2 -1)2=
W’ < n/2 -1=
Shift one time for each vertex
Drawing of G’
![Page 159: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/159.jpg)
G’
G”
W < n/2 -1
W/2
W/2
1
=
Step 5: Combine the drawing of G’ and G”
![Page 160: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/160.jpg)
Step 5: Combine the drawing of G’ and G”
G’
G”
W < n/2 -1
W/2
W/2
1
=
| 辺の傾き |>1
| 外周上の辺の傾き | < 1 =
![Page 161: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/161.jpg)
| 外周上の辺の傾き | < 1 =
G’
G”
| 辺の傾き |>1
Step 5: Combine the drawing of G’ and G”
![Page 162: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/162.jpg)
x- 単調
G’
G”
Step 5: Combine the drawing of G’ and G”
![Page 163: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/163.jpg)
G’
G”
W < n/2 -1
W/2
W/2
1
=
G
W < n/2 -1
H< n/2
=
=
Step 5: Combine the drawing of G’ and G”
![Page 164: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/164.jpg)
W = n/2 -1
H = n/2
Output: a straight line grid drawing Grid Size :
(W +H < n-1) W < n/2 -1, H < n/2
== =
Area: W x H < = ( )n -12
2 W < n/2 -1
H < n/2
=
=
Input: 4-connected plane graph G
The algorithm isbest possible
Miura et al. ‘01
![Page 165: Drawing Plane Graphs](https://reader035.vdocuments.site/reader035/viewer/2022062518/568148a9550346895db5bcdc/html5/thumbnails/165.jpg)
ObjectiveObjective
To minimize the number of bends in an orthogonal drawing.
an orthogonal drawing with the minimum number of bends.
4 bends
de
a b c
fg
hi
9 bends
a bc
def
gh
i