WADS 2013

WADS 2013 August 12, 2013

Plane 3-trees: Embeddability & Approximation

Department of Computer ScienceUniversity of Manitoba

Stephane Durocher Debajyoti Mondal

Point-Set Embeddings

A plane graph G

1WADS 2013 August 12, 2013

A point set S

Input

a

b

c

de fg

hi

b

c

de fg

hi

Point-Set Embeddings

An embedding of G on S

2WADS 2013 August 12, 2013

bh

i

cdea

fg

Output

A plane graph G A point set S

Input

a

b

c

de fg

hi

b

c

de fg

hi

Not Always Embeddable

A plane graph G

3WADS 2013 August 12, 2013

A point set S

Inputa

b

e

b

cd

f

Not Always Embeddable

A plane graph G

4WADS 2013 August 12, 2013

A point set S

Inputa

b

e

b

cd

f

b

a

c b

a

c b

a

c

d

dd

Plane 3-treesNishat et al.(2010)O(n2)Moosa & Rahman(2011) O(n4/3 + ɛ) This Presentation O(n lg3 n)

2-bend embeddability 1/ √n Approximation

Previous Results

5WADS 2013 August 12, 2013

Outerplanar graphsGritzmann et al. (1991), Castañeda & Urrutia (1996) O(n2), Bose (2002) O(n lg3n)

NP-completeCabello (2006 ) 2-outerplanar, Nishat et al. (2011 ) partial 3-tree, Durocher & M.(2012 ) 3-connected,Biedl &Vatshelle (2012 )

3-connected, fixed treewidth

Not Embeddable

Plane 3-Trees

6WADS 2013 August 12, 2013

a

b

c

d

e

f

A plane 3-tree

Plane 3-Trees

7WADS 2013 August 12, 2013

a

b

c

d

e

f

A plane 3-tree

a

b

c

Insert e

a

b

c

d

a

b

c

d

e

Insert d

Insert f

Properties of Plane 3-Trees

8WADS 2013 August 12, 2013

Plane 3-Tree Plane 3-Tree

Plane 3-Tree Representative Vertex

a

b

c

d

e

fb

ce

d

a

d

b f

d

c

a

How to Solve the Problem…

9WADS 2013 August 12, 2013

a

b d

Convex Hull

f

h

A Plane 3-Tree G

A Point-Set S

g

ce

How to Solve the Problem…

10WADS 2013 August 12, 2013

b

c

d

e

a

fgh

c

a

b

A Plane 3-Tree G

A Point-Set S

a

b

c

Valid Mapping of the Representative Vertex

11WADS 2013 August 12, 2013

d

c

b

n1=4

n2=4

n3=5 3

6

4a

c

d

e

a

fgh

b


12WADS 2013 August 12, 2013

b

c

d

e

a a

c

bd

4

4

5

fgh

n1=4

n2=4

n3=5

Nishat et al. (2010): The mapping of the representative vertex is unique.


13WADS 2013 August 12, 2013

b

c

d

e

a a

c

bd

4

4

5

fgh

n1=4

n2=4

n3=5

T(n) = T(n1) + T(n2) + T(n3) + O(n2) = O(n3)

Improvements

14WADS 2013 August 12, 2013

T(n) = T(n1) + T(n2) + T(n3) + O(n2) = O(n3)

T(n) = T(n1) + T(n2) + T(n3) + O(n) = O(n2)

T(n) = T(n1) + T(n2) + T(n3) + min{n1, n2, n3} . n1/3+ ɛ

= O(n4/3+ ɛ)

T(n) = T(n1) + T(n2) + T(n3) + min{n1+n2, n2+n3, n3+n1}. lg2 n

= O(n lg3n)

Moosa and Rahman(COCOON 2011)

Nishat et al. (GD 2010)

This Presentation

New Techniques

15WADS 2013 August 12, 2013

W. Steiger and I. Streinu (1998)Given a triangular set S of n points in general position, in O(n) time one can construct a new point m such that the sub-triangles contain prescribed number of points of S.

(i+j+k) -3 = 10 = n

m

x

zy

i = 4

j = 5

k = 4

(i+j+k) -3 = 10 = n

m'

x

zy

i = 4

j = 5

k = 4

New Techniques

16WADS 2013 August 12, 2013

The partition of the interior points into subtriangles is unique !

(i+j+k) -3 = 10 = n

m

x

zy

i = 4

j = 5

k = 4

(i+j+k) -3 = 10 = n

m'

x

zy

i = 4

j = 5

k = 4

New Techniques

17WADS 2013 August 12, 2013

b

c

d

e

a a

c

bm

n1 - 1

n2+ 1

n3 - 1

fgh

n1=4

n2=4

n3=5

New Techniques

18WADS 2013 August 12, 2013

b

c

d

e

a a

c

bm

n1 - 1

n2+ 1

n3 - 1

fgh

n1=4

n2=4

n3=5

d

New Techniques [O(nlg3n) time?]

19WADS 2013 August 12, 2013

b

c

d

e

a a

c

bd

n1

n2

n3

fgh

n1=4

n2=4

n3=5

T(n) = T(n1) + T(n2) + T(n3) + O(n) = O(n2)

Valid Mapping in O((n2+n3) lg2n) time

20WADS 2013 August 12, 2013

a

c

b

n1

n2

n3

a

c

b

n1-1

n3-1

n2-1u

v

n3 ≤ n2 ≤ n1

The representative vertex must lie inside the green region.


21WADS 2013 August 12, 2013

a

c

b

n1

n2

n3

a

c

b

n3-1

n3-1

n3-1u

v

n3 ≤ n2 ≤ n1

The green region contains O(n3) points.The representative vertex and its incident edges must lie inside the green region.

r

s


22WADS 2013 August 12, 2013

a

c

b

n1

n2

n3

a

c

bu

v

n3 ≤ n2 ≤ n1

The green region contains O(n3) points.The representative vertex and its incident edges must lie inside the green region.


23WADS 2013 August 12, 2013

a

c

b

n1

n2

n3

a

c

bu

v

Finding a valid mapping in S with partition (n1, n2, n3)

Finding a valid mapping in S/

with partition (n1-x1, n2-x2, n3) or, (n3+1, n3+1, n3)

x1

x2

Point-set Embedding in O(nlg3n) time

24WADS 2013 August 12, 2013

Select O(n2+n3) candidate points in O((n2+n3) lg2n) time

Find the required mapping in the reduced point set in O(n3) time

T(n) = T(n1) + T(n2) + T(n3) + O(min{n1+n2, n2+n3, n3+n1}. lg2 n ) = O(n lg3n)

2-Bend Point-Set Embeddings

20WADS 2013 August 12, 2013

a

b c

df

p

e

A 2-bend point-set embedding of G on S

Output

A plane 3-tree G A point set S

Input

M. Kaufmann and R. Wiese (2002)

Every plane graph admits a 2-bend point set embedding with O(W 3) area on any set of n points in general position.

2-Bend Point-Set Embeddings

21WADS 2013 August 12, 2013

a

b

c

a

b

c

d

e

f

a

b

c

de

fa

b

cd

a

b c

e

d

f

Approximable with Factor 1/√n

22WADS 2013 August 12, 2013

a

b c

e

d

f

A plane 3-tree G S


23WADS 2013 August 12, 2013

a

b c

e

d

f

S

S(Γ ) = 3

A plane 3-tree G

p1

p2

p3

p4

p5

p6

p1 p3

p5


24WADS 2013 August 12, 2013

a

b c

e

d

f

S

S(Γ *) = 4

S(Γ )Approximation factor = __________

S(Γ * )

S(Γ ) = 3

A plane 3-tree G

p1

p2

p3

p4

p5

p6

p1 p3

p5

p1

p3

p5

p6

Future Research

25WADS 2013 August 12, 2013

Variable embedding: Is there a subquadratic-time algorithm for testing point-set embeddability of plane 3-trees in variable embedding setting?

1-Bend Point-Set Embeddability: Is it always possible to find 1-bend point set embeddings for plane 3-trees?

Approximation: Is it possible to approximate point-set embeddability of plane 3-trees within a constant factor ?

Thank You..

Documents

WADS 2013