drawing plane graphs takao nishizeki tohoku university

Drawing Plane Graphs

Takao Nishizeki

Tohoku University

US PresidentCalifornia Governor

What is the common feature?

Graphs and Graph Drawings

A diagram of a computer network

ATM-SW

ATM-HUBATM-RT

ATM-RT

ATM-HUB

ATM-RT

ATM-HUB

STATIONSTATION

STATION

STATIONTPDDI

STATIONATM-HUB

STATION

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

Nice drawing

Ancient beauty

Modernbeauty

The drawing should satisfy some criterion arising from the application point of view.

not suitable for single layered PCB

suitable for single layered PCB

Diagram of an electronic circuit

Wire crossings

Drawings of Plane Graphs

Convex drawing

Straight line drawing

Drawings of Plane Graphs

Rectangular drawing

Orthogonal drawing

Box-rectangular drawing

Planar Graph Drawing

Takao NishizekiMd. Saidur Rahman

http://www.nishizeki.ecei.tohoku.ac.jp/nszk/saidur/gdbook.html

Straight Line Drawing

Plane graph

Plane graphStraight line drawing

Each vertex is drawn as a point.

Each edge is drawn as a single straight line segment.

Every plane graph has a straight line drawing.

Wagner ’36 Fary ’48Polynomial-time algorithm

Straight Line Grid Drawing

In a straight line grid drawing each vertex is drawn on a grid point.

Plane graph

Straight line grid drawing.

Wagner ’36 Fary ’48

Plane graph

Grid-size is not polynomial of the number of vertices n

Plane graph

de Fraysseix et al. ’9022nHW

Straight Line Grid Drawing

Plane graph

de Fraysseix et al. ’9022nHW

Chrobak and Payne ’95

Linear algorithm

Schnyder ’90

2nHW Upper bound

What is the minimum size of a grid required for a straight line drawing?

Lower Bound

A restricted class of plane graphs may have more compact grid drawing.

Triangulated plane graph

3-connected graph

4-connected ?

not 4-connected disconnected

How much area is required for 4-connected plane graphs?

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

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

The algorithm of Miura et al. isbest possible

|slope|>1|slope|>1|slope|>1

Main idea

Step2: Divide G into two halves G’ and G”

G’1 2

n-1=17 n=18

W < n/2 -1=

Step3 and 4 : Draw G’ andG” in isosceles right-angledtriangles

n/2 -1

Step5: Combine the drawings of G’ and G”

Triangulate all inner faces

n-1=17 n=18

Step1: find a 4-canonical ordering

|slope|<1=

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.

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

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

Convex Grid Drawing

Chrobak and Kant ’97

Input: 3-connected graph

Output: convex grid drawing

Grid Size

Input : 4-connected plane graph

Miura et al. 2000

Output: Convex grid drawing

Half-perimeter

Convex Grid Drawing

Grid Size

Area < 1/4=

3-connected graph 4-connected graph

Chrobak and Kant ’97 Miura et al. 2000

The algorithm of Miura et al. isbest possible

1: 4-canonical decomposition1 2

1011 12

131415

1718 19

16 1718 19

9 107 86

3 4 54: Decide y-coordinates

1 2 Time complexity: O(n)

O(n)[NRN97]2: Find paths

3: Decide x-coordinates

Main idea

VLSI FloorplanningVLSI Floorplanning

Interconnection graph

Interconnection graph VLSI floorplan

Dual-like graph

Dual-like graph Add four corners

Rectangular drawing

Rectangular Drawings

Plane graph G of 3

Rectangular drawing of GPlane graph G of 3

Input Output

corner

Each vertex is drawn as a point.Input Output

corner

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

corner

Each face is drawn as a rectangle.

corner

Not every plane graph has a rectangular drawing.

Thomassen ’84,

Necessary and sufficient condition

Rahman, Nakano and Nishizeki ’98

Linear-time algorithms

Linear algorithm for the case where corners are not designated in advance

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.

2-legged cycles

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

Outline of the Algorithm of Rahman et al.

Bad cycle

Partition-pair

Partition-pair Splitting

Partition-pair

Rectangular drawing

Partition-pair

Rectangular drawing

Time complexity

Miura, Haga, N. ’03, Working paper

Rectangular drawing of plane graph G with Δ ≤ 4

perfect matching in Gd

Gd perfect matching

G )( 5.1nO

Rectangular drawing

Unwanted adjacency

Not desirable for MCM floorplanning andfor some architectural floorplanning.

MCM FloorplanningMCM Floorplanning SherwaniSherwani

Architectural FloorplanningArchitectural Floorplanning Munemoto, Katoh, ImamuraMunemoto, Katoh, Imamura

MCM FloorplanningMCM FloorplanningArchitectural FloorplanningArchitectural Floorplanning

Dual-like graph

Box-Rectangular drawing

Dual-like graph

dead space

Box-Rectangular Drawing

InputPlane multigraph 　 G

Input OutputPlane multigraph 　 G

cdBox-rectangular drawing

• Each vertex is drawn as a rectangle.

Plane multigraph 　 G

Output

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

segment.

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.

Output

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.

Algorithm of Rahman et al.

Main Idea: Reduction to a rectangular drawing problem

Outline

Replace each vertex of degree four or more by a cycle

Outline

G linear time[RNN98a]

Rectangular drawing

Outline

Time complexity

Orthogonal Drawings

plane graph Ga b

Orthogonal Drawings

Input Output plane graph G

Orthogonal Drawings

Each edge is drawn as an alternating sequence of horizontal and vertical line segments.

Orthogonal Drawings

Each edge is drawn as an alternating sequence of horizontal and vertical line segments.Each vertex is drawn as a point.

Applications

Entity-relationship diagramsFlow diagrams

Applications

Circuit schematics

Minimization of bends reduces the number of “vias” or “throughholes,” and hence reduces VLSI fabrication costs.

plane graph Ga b

iObjectiveObjective

4 bends

minimum number of bends.

9 bend

To minimize the number of bends in an orthogonal drawing.

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.

Bend-Minimum Orthogonal Drawing

Linear

Rahman and Nishizeki ’02

Linear for plane graph of Δ≤3

Outline of the algorithm of [RNN99]

)( 1CG

)( 2CG

orthogonal drawings

rectangular drawing of G

orthogonal drawing of G

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

nHW for every planar graph?

Linear algorithm for bend-minimal orthogonal drawings of plane graphs of Δ ≤4.

ISAAC in Sendai, Tohoku

Probably 2007

SendaiKyoto

Properties of a drawing of G(C)

minimum number of bendsthe six open halflines are free

orthogonal drawing of G(C)

Main idea

　　 4-canonical decomposition1 2

1011 12

131415

1718 19

Main idea

1011 12

131415

1718 19

16 1718 19

9 107 86

1 21 2

Add a group of vertices one by one.

Main idea

1011 12

131415

1718 19

16 1718 19

9 107 86

1:4-canonical decomposition1 2

1011 12

131415

1718 19

16 1718 19

9 107 86

3 4 54: Decide y-coordinates

1 2 Time complexity: O(n)

O(n)[NRN97]2: Find paths

3: Decide x-coordinates

Main idea

4-canonical decomposition[NRN97]

(a generalization of st-numbering)

4-canonical decomposition[NRN97](a generalization of st-numbering)

1213 1415

1718 19

20 n = 21

Orthogonal Drawings

Each edge is drawn as an alternating sequence of horizontal and vertical line segments.

ApplicationsApplicationsCircuit schematics, Data-flow diagrams, Entity-relationship diagrams [T87, BK97].

ObjectiveObjectiveTo minimize the number of bends in an orthogonal drawing.

an orthogonal drawing with the minimum number of bends.

9 bends

4 bends

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]

Outline of the algorithm of [RNN99]

)( 1CG

)( 2CG

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

orthogonal drawing of G(C)

Each face is drawn as a rectangle.

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

Rectangular Drawing Algorithm of [RNN98]

Splitting

Partition-pairInput

rectangular drawing of each subgraph

Partition-pair

Modification of drawingspatching

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.

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.

Application VLSI floorplanning

Interconnection Graph Floor plan

Dual-like graph Insertion of four corners

Application VLSI floorplanning

Interconnection Graph Floor plan

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

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

G’’

G’ |V’|= n/2

|V”|= n/2

Step 2: 　 Divide G into G’and G”

n-1=17 n=18

n/2 =9

Step 3: 　 Draw G’ in an isosceles right-angled triangle

G’ |V’|= n/2

|slope| < 1 =

|slope| > 1

W < n/2 -1=

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

Step 3: 　 Draw G’ in an isosceles right-angled triangle

shift and put k

各点の座標の決め方

G3Initialization ：

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 = =

The drawing of the path passing through all k’s neighbors is “weakly convex”

“weakly convex”

not “weakly convex”

Case 1 ： k has exactly one highest neighbor

shift and put k

The rightmost neighbor of k is higher than the leftmost neighbor

shift and put k

The rightmost neighbor of k is higher than the leftmost neighbor

Case 2 ： k has exactly two or more highest neighbor

Drawing of G’

H’ <W’/2=( n/2 -1)2=

W’ < n/2 -1=

Shift one time for each vertex

Drawing of G’

W < n/2 -1

Step 5: Combine the drawing of G’ and G”

W < n/2 -1

| 辺の傾き |>1

| 外周上の辺の傾き | < 1 =

| 辺の傾き |>1

x- 単調

W < n/2 -1

H< n/2

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

ObjectiveObjective

To minimize the number of bends in an orthogonal drawing.

an orthogonal drawing with the minimum number of bends.

4 bends

9 bends

drawing plane graphs takao nishizeki tohoku university

connected plane graph

single straight line

plane nicelya convex

compact grid drawing

grid point

connected plane graphs

convex grid drawingchrobak

graph drawingsa diagram

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