wads 2013 august 12, 2013 department of computer science university of manitoba stephane durocher...
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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
b
h
i
cde
a
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-trees
Nishat 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 graphs
Gritzmann 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
f
gh
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
4
a
c
d
e
a
f
gh
b
Valid Mapping of the Representative Vertex
12WADS 2013 August 12, 2013
b
c
d
e
a a
c
bd
4
4
5
f
gh
n1=4
n2=4
n3=5
Nishat et al. (2010): The mapping of the representative vertex is unique.
Valid Mapping of the Representative Vertex
13WADS 2013 August 12, 2013
b
c
d
e
a a
c
bd
4
4
5
f
gh
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
f
gh
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
f
gh
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
f
gh
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.
Valid Mapping in O((n2+n3) lg2n) time
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
Valid Mapping in O((n2+n3) lg2n) time
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.
Valid Mapping in O((n2+n3) lg2n) time
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
d
fp
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
Approximable with Factor 1/√n
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
p1p3
p5
Approximable with Factor 1/√n
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
p1p3
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..
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