Orthogonal Drawings of Series-Parallel Graphs Joint work with Xiao Zhou by Tohoku University Takao Nishizeki

Orthogonal Drawings of Series-Parallel Graphs

Joint work with Xiao Zhou

by

Tohoku University

Takao Nishizeki


with Minimum Bends

Xiao Zhou and Takao Nishizeki

by

Tohoku University

1. each vertex is mapped to a point2. each edge is drawn as a sequence of alternate horizontal and

vertical line segments3. any two edges don’t cross except at their common ends

planar graph orthogonal drawings


with Minimum Bends


by

Tohoku University




bend

one bend


with Minimum Bends


by

Tohoku University



planar graph orthogonal drawingscrossing


with Minimum Bends


by

Tohoku University




one bend no bend

another embedding

bend

bend

An orthogonal drawing of a planar graph G is optimal if it has the minimum # of bends among all possible orthogonal drawings of G.

orthogonal drawings

one bend no bend

Optimal orthogonal drawing

optimal

planar graph

bend via-hole, through-hole VLSI design

Example

4 bends

Orthogonal Drawings

drawing

bendbendbendbend

Example

4 bends

Orthogonal Drawings

drawing

bendbendbendbend

Optimal ? No

Example

embedding

Orthogonal Drawings

drawing

bendbendbendbend

flip

4 bends

Example

embedding

flip

4 bends

Orthogonal Drawings

drawing

bendbendbendbend

Example

embedding

flip

4 bends

Orthogonal Drawings

drawing

bendbendbendbend

Example

embedding

flip

4 bends

Orthogonal Drawings

drawing

bendbendbendbend

Example

embedding

4 bends

Orthogonal Drawings

drawing

bendbendbendbend

Example

drawing

embedding

0 bend

optimal

4 bends

Orthogonal Drawings

drawing

bendbendbendbend

Orthogonal Drawings

drawing

bendbendbendbend

4 bends

Example

drawing

embedding

0 bend

optimal

Given a planar graph

Wish to findan optimal orthogonal drawing

Find an optimal orthogonal drawing of a given planar graph.

Problem

wheredegree of vertex v: # of edges incident to vΔ ： max degree

Δ=3, n=7

Known results

The problem:NP-complete for planar graphs of Δ≦4

A. Garg, R. Tamassia, 2001

If Δ≧5, then no orthogonal drawing.d(v)=2

d(v)=3

2

2 2

2

323

Known results



D. Battista, et al. 1998If Δ≦3, O(n5logn) time

where n: # of vertices

Δ=3, n=7

Known results



If Δ≦3, O(n5logn) time D. Battista, et al. 1998

If Δ≦4, O(n4) timeIf Δ≦3, O(n3) time

D. Battista, et al. 1998

D. Battista, et al. 1998

For biconnected series-parallel graphs

Our results

If Δ≦3, O(n) time our result

much simpler and faster than the known algorithms

For series-parallel graphs

Series-Parallel Graphs

(a)

if are SP graphs,

is a SP graph.

then are SP graphs

A SP graph is recursively defined as follows:

(b)

a single edge


(a)

if are SP graphs,

is a SP graph.

then are SP graphs


(b)

a single edge

parallel-connection


(a)

if are SP graphs,

is a SP graph.

then are SP graphs


(b)

a single edge

series-connection parallel-connection


(a)

if are SP graphs,

is a SP graph.

then are SP graphs


(b)

a single edge


(a)

if are SP graphs,

is a SP graph.

then are SP graphs


(b)

a single edge


Example

SP graph


Example

SP graphseries-connection


Example

SP graphparallel-connection


Example

SP graphSeries-connection


Example

SP graphParallel-connection


Example

Biconnected SP graphs

SP graphbiconnected

Biconnected graphs G:

G – v is connected for each vertex v.v v


Example

SP graphbiconnected

Biconnected graphs G:

G – v is connected for each vertex v.

not biconnected

v

Biconnected SP graphs

cut vertex


Example

SP graphsbiconnected

(Our Main Idea)Lemma 1

Every biconnected SP graph G of Δ≦3 has one of thefollowing three substructures: (a) a diamond C (b) two adjacent vertices u and v s.t. d(u)=d(v)=2 (c) a triangle K3.

(a)

u v

(b)

K3

(c)

2

3

2

3 2 22

3 3

(Our Main Idea)Lemma 1

Every biconnected SP graph G of Δ≦3 has one of thefollowing three substructures: (a) a diamond C (b) two adjacent vertices u and v s.t. d(u)=d(v)=2 (c) a triangle K3.

(a)

u v K3

(b) (c)

2

3

2

3 2 22

3 3

Algorithm(G)

Let G be a biconnected SP graph of Δ≦3.Case (a): ∃ a diamond , Recursively find an optimal drawing.

　Case (b): ∃

　 Decompose to smaller subgraphs in series or

parallel, and iteratively find an optimal drawingCase (c): ∃

　　　　 Similar as Case (b).

Let G be a biconnected SP graph of Δ≦3.Case (a): ∃ a diamond , 　

Algorithm(G)

Algorithm( ),

Recur G to a smaller one 　　　　

find

optimal optimal

contractReturn

expand

expand


Algorithm(G)

Algorithm( ), find

optimal optimal

contractReturn

expand

expandcontract contract

Examplecontract

drawing

delete edge


Algorithm(G)

Algorithm( ), find

contractReturn

expand

Case (b): ∃

deg=1 deg=1

2-legged SP

Find an opt & U-shape

delete edge

Case (b): ∃

terminals are drawn on the outer face;the drawing except terminals doesn’t intersect the north side

Our Main Idea

Definition of I- , L- and U-shaped drawings

I-shape L-shape U-shape

2-legged SP graph

s tU-shape U-shape

Our Main Idea

2-legged SP graph

s t


L-shape U-shape

terminals are drawn on the outer face;the drawing except terminals intersects neither the north side nor the south side I-shape

I-shape

I-shape

terminals are drawn on the outer face;the drawing except terminals intersects neither the north side nor the east side

Our Main Idea



2-legged SP graph

s tL-shapeL-shape

Lemma 2

Every 2-legged SP graph without diamond has optimal I-, L- and U-shaped drawings

I-shape

optimal

bend

L-shape

optimal

bend

U-shape

optimal

bend

Δ ≦ 3

deg=1 deg=1

2-legged SP


parallel connectionseries connection

delete edge

Case (b): ∃

decompose

opt & L-shape opt & L-shapeopt & U-shapeopt & U-shapeopt & L-shape


delete edge

Case (b): ∃

opt & U-shapeopt & I-shapeopt & U-shape

opt & U-shape

opt &I-shape

opt & U-shape


delete edge

Case (b): ∃

opt & U-shapeopt & I-shapeopt & U-shape

opt & U-shape

opt & I-shape


delete edge

Return an optimal drawing

extend

Case (b): ∃

Algorithm( ),

Algorithm(G)

u vu v

Case (c):∃ a complete graph K3. opt & U-shape

Return optimal drawing

bend

expand


contraction

Case (b): ∃

An optimal orthogonal drawing of a biconnected SP graph G of Δ≦3 can be found in linear time.

Theorem 1


Theorem 1Our algorithm works well even if G is not biconnected.

find three types of drawings

best one of

Optimal drawing

=

bendbend

bend bendbend

bend bendbendbend

4 bends

2 bends

3 bends

An optimal orthogonal drawing of a SP graph G of Δ≦3 can be found in linear time.

Theorem 1Conclusions

Conclusions

bend(G) ≦ n/3 for biconnected SP graphs G of Δ≦3

Grid size ≦ 8n/9

=width + height

width

height

Optimal orthogonal drawings ?


planar graph

a 2-bend orthogonal drawing


1-connected SP graph

not optimal

one bend

bend

not optimal

one bend

bend

Is this optimal ?

∃a one-bend orthogonal drawing ?


one bendno bendoptimal not optimal

bend



Yes

one bend

one bendno bendoptimal not optimal

bendbend


Is this optimal ?

∃a 0-bend orthogonal drawing ?

orthogonal drawings


0 bend

optimal



0 bend


optimal

0 bend

optimal


orthogonal drawings



No

optimal

no bend

optimal

no bend

crossing crossing



one bendoptimal

bendbend

two bends

bend



Theorem 1Conclusions

Our algorithm works well even if G is not biconnected.

min # of bends

min # of bends=


Conclusions

bend(G) ≦ (n+4)/3 for SP graphs G of Δ≦3

Our algorithm works well even if G is not biconnected.

Best possible

bend(G)= (n – 8)/3 + 4 = (n +4)/3

bend

bend bend

bend

bendbendbend

Is there an O(n)-time algorithm to find an optimal orthogonal drawing of G ?

For series-parallel graphs G with Δ=4

open

s t s t s t

s t s t

s toptimal I-shapeoptimal L-shape

no optimal U-shape

Optimal drawing

Our Main Idea

2-legged SP graph

A SP graph G is 2-legged if n(G) 3 and ≧ d(s)=d(t)=1 for the terminals s and t.

s t

Our Main Idea

2-legged SP graph


s t


L-shape U-shape

terminals are drawn on the outer face;the drawing except terminals intersects neither the north side nor the south side I-shape

I-shape

I-shape

terminals are drawn on the outer face;the drawing except terminals intersects neither the north side nor the east side

Our Main Idea

2-legged SP graph




s

L-shape

tL-shapeL-shape

terminals are drawn on the outer face;the drawing except terminals doesn’t intersect the north side

Our Main Idea

2-legged SP graph


st



U-shape tnot U-shapeU-shape

Lemma 2

The following (a) and (b) hold for a 2-legged SP graph Gof Δ 3 unless ≦ G has a diamond: (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time.

optimal U-shaped drawings

optimal I-shaped drawings

Lemma 2

The following (a) and (b) hold for a 2-legged SP graph Gof Δ 3 unless ≦ G has a diamond: (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time.

optimal

I-shape L-shape

optimal

U-shape

optimal

Definition of Diamond Graph

(a)

if are diamond graphs,

is a diamond graph.

then is a diamond graph

A Diamond graph is recursively defined as follows:

(b)

a path with three vertices

Definition of Diamond Graph

(a)

if are diamond graphs,

is a diamond graph.


A Diamond graph is recursively defined as follows:

(b)

a path with three edges

Diamond Graph

(a) is a diamond graph.

If and are diamond graphs, (b)


Diamond Graph

(a) is a diamond graph.

If and are diamond graphs, (b)


If G is a diamond graph, then (a) G has both a no-bend I-shaped drawing and a no-bend L-shaped drawing

(b) every no-bend drawing is either I-shaped or L-shaped.

Lemma 1

I-shape

L-shape

∃no-bend U-shaped drawing ?

NO

1-bend U-shaped drawing

Lemma 2

The following (a) and (b) hold for a 2-legged SP graph Gof Δ≦3 unless G is a diamond graph (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time.

I-shape

L-shape

diamond graph

no-bend drawing I-shapeL-shape

not a diamond graph

U-shape

Known results

In the fixed embedding setting:

Min-cost flow problem

n : # of verticesn : # of verticesFor plane graph:

O(n2logn) time R. Tamassia, 1987

Known results

In the fixed embedding setting:

improved

n : # of verticesn : # of verticesFor plane graph:

O(n2logn) time R. Tamassia, 1987O(n7/4 logn) time A. Garg, R. Tamassia, 1997

Known results

In the fixed embedding setting:For plane graph:


In the variable embedding setting:NP-complete for planar graphs of Δ≦4


Known results

In the fixed embedding setting:For plane graph:


In the variable embedding setting:NP-complete for planar graphs of Δ≦4


If Δ≦3, O(n5logn) time D. Battista, et al. 1998

Lemma 3

Every biconnected SP graph G of Δ≦3 has one of thefollowing three substructures: (a) a diamond C (b) two adjacent vertices u and v s.t. d(u)=d(v)=2 (c) a complete graph K3.

(a) G

Proof

G G

bend(G) bend(≦ G )

Given an optimal drawing of G

Proof

G G

bend(G)≧bend(G )

Given an optimal drawing of G

Case 1: bend( ) = 0

Case 2: bend( ) = 1

Case 3: bend( ) 2≧omitted

Lemma 1

Every biconnected SP graph G of Δ≦3 has one of thefollowing three substructures: (a) a diamond (b) two adjacent vertices u and v s.t. d(u)=d(v)=2 (c) a complete graph K3.

u v

G

u v

G G

bend(G)=bend(G )

(Our Main Idea)

No diamonds

∃an optimal U-shape drawing

G

Lemma 2

For each SP graph G If n ≧ 4, Δ≦3 and d(s)=d(t)=1, then ∃an optimal U-shape drawing

u v

G

u v

G

bend(G)=bend(G )

No diamonds


s t

optimal U-shaped drawing

s t

G

Lemma 1

Every biconnected SP graph G of Δ≦3 has one of thefollowing three substructures: (a) a diamond (b) two adjacent vertices u and v s.t. d(u)=d(v)=2 (c) a complete graph K3.

K3

G G

bend(G)=bend(G )+1

(Our Main Idea)

No diamonds G


Lemma 2

K3

G G G

bend(G)=bend(G )+1

No diamonds G


For each SP graph G If n ≧ 4, Δ≦3 and d(s)=d(t)=1, then ∃an optimal U-shape drawing

s t

optimal U-shaped drawing

s t

Optimal orthogonal drawings ?


1-connected planar graphnon

deg=1 deg=1

2-leg SP


U-shape not U-shapeU-shape

parallel connectionseries connection

delete edge

Case (b): ∃

Documents

Orthogonal Drawings of Series-Parallel Graphs Joint work with Xiao Zhou by Tohoku University Takao Nishizeki