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GD 2011September 21,
2011
Embedding Plane 3-Trees in R2 and R3
Stephane Durocher
Debajyoti Mondal
Md. Saidur RahmanSue Whitesides
Rahnuma Islam Nishat
Dept. of Computer Science and Engg.Bangladesh University of
Engineering and Technology
Department of Computer ScienceUniversity of Victoria
Department of Computer ScienceUniversity of Manitoba
GD 2011September 21,
2011
Point-Set Embeddings
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de fg
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A plane graph G A point set P
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GD 2011September 21,
2011
Point-Set Embeddings
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de fg
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A plane graph G An embedding of G on P
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de fg
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GD 2011September 21,
2011
Previous Results
Gritzmann et al. (1991),Castañeda and Urrutia (1996) Outerplanar graphs O(n2)
Bose (2002) Outerplanar graphs O(n lg 3n)
Cabello (2006) Biconnected 2-outerplanar graphs NP-complete
Nishat et al. (2010)Plane 3-trees
Partial plane 3-treesO(n2),
NP-complete
Moosa et al. (2011) Plane 3-trees O(n4/3 + ɛ log n)
This PresentationPlane 3-trees, O(n4/3 + ɛ )-time algorithm in R2,
NP-complete in R3, when a mapping for the outervertices is prespecified
Reference Graph Class Time complexity
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GD 2011September 21,
2011
Plane 3-Trees
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hi
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A plane 3-tree G
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oa
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A construction for G
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GD 2011September 21,
2011
a
b
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de
fg
hi
j
k
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mn
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A plane 3-tree G
fg
hi
j
k
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mn
oa
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The representative vertex of G
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le
A plane 3-tree
A plane 3-tree
A construction for G
Properties of Plane 3-trees
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mn
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A plane 3-tree
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GD 2011September 21,
2011
a
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Convex Hull
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General Idea of the Algorithm
A plane 3-tree G A point set P
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GD 2011September 21,
2011
General Idea of the Algorithm
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a c
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afgh
A plane 3-tree G A point set P
We can map the outervertices in Six different ways.
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GD 2011September 21,
2011
General Idea of the Algorithm
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a a
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Find a valid mapping for the representative vertex.
fgh
Valid mapping??
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fgh
n1 = 1
n2 = 1
n3 = 2
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GD 2011September 21,
2011
a
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bd
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1
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Find a valid mapping for the representative vertex.
General Idea of the Algorithm
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a
fgh
a
b
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fgh
n1 = 1
n2 = 1
n3 = 2
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GD 2011September 21,
2011
General Idea of the Algorithm
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a a
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bd
fgh
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hf
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Find a valid mapping for the representative vertex recursively.
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fgh
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GD 2011September 21,
2011
How fast can we find a valid mapping for the representative vertex,
if such a mapping exists?
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GD 2011September 21,
2011
Finding a Valid Mapping
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Representative vertex cannot be mapped in the shaded regions.
At most min{n1, n2, n3}+1 points in the white region are candidates.
fgh
a
b
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fgh
Assume that n1≤ min{n2,n3}.
a
n3 = 2
n2 = 1
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n1 = 1
n2 = 1
n3 = 2
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GD 2011September 21,
2011
Finding a Valid Mapping
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fgh
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fgh
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Choose a random point z ϵΔabcWe need n3 points in this region
How do we select the shaded regions?
n1 = 1
n2 = 1
n3 = 2
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GD 2011September 21,
2011
Finding a Valid Mapping
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fgh
a
b
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fgh
How do we select the shaded regions?
z
Choose a random point z ϵΔabcWe need n3 points in this regionn1 = 1
n2 = 1
n3 = 2
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GD 2011September 21,
2011
How do we select the shaded regions?
Finding a Valid Mapping
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a
fgh
a
b
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fgh
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a
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n3 = 2
n2 = 1
n1 = 1
n2 = 1
n3 = 2
Selecting the shaded regions takes O(tn log n) expected time.
At most min{n1, n2, n3}+1 points in the white region are candidates.
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GD 2011September 21,
2011
a
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bd
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Find a valid mapping in fn = O(tn log n) + O(tn min{n1, n2, n3}) time.
T(n) = T(n1)+ T(n2)+ T(n3)+ fn
= O(n4/3 + ɛ ), for any ɛ > 0, using Chazelle’s DS.
Time Complexity
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fgh
a
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fgh
n1 = 1
n2 = 1
n3 = 2
GD 2011September 21,
2011
Extension to R3 ?
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a c
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afgh
A plane 3-tree G A point set P and a prespecified mapping for the outervertices of G
The problem is NP-hard when the points are in R3. Remove the general position assumption.
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GD 2011September 21,
2011
3-Partition
Instance: A set of 3m nonzero positive integers S = {a1, a2,...,a3m}
and an integer B > 0, where a1+a2+...+a3m = mB and
B/4 <ai <B/2,1 ≤ i ≤ 3m.
Question: Can S be partitioned into m subsets S1,S2,...,Sm such that
|S1| =|S2| = ... = |Sm| =3 and the sum of the integers in each
subset is equal to B?
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 , 8 < ai < 16
S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}
Example:
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GD 2011September 21,
2011
3-Partition PSE in R3
a 1a 2
a 3mBB
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Z
Xa
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ca18
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P
GD 2011September 21,
2011
Idea of the Hardness Proof
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32
S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}
Example:
B B B B
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x
GD 2011September 21,
2011
Idea of the Hardness Proof
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32
S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}
Example:
B B B B9
10
11
a1
a2
a|S| A fan
A divider
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x
GD 2011September 21,
2011
Idea of the Hardness Proof
S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32
S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11}
Example:
20
x
{10, 10, 12} { 9, 11, 12} { 9, 9,14} { 10, 11,11}
GD 2011September 21,
2011
Idea of the Hardness Proof
a 1a 2
a 3mBB
Y
b
c
Z
Xa
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ca21
G
P
GD 2011September 21,
2011
Idea of the Hardness Proof
BB
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ca
A fan
A dividera
A spine vertex
22G
P
GD 2011September 21,
2011
Idea of the Hardness Proof
BB
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ca
A fan
A dividera
A spine vertex
23G
P
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GD 2011September 21,
2011
Idea of the Hardness Proof
BB
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b
ca
EdgeCrossings?
a
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A spine vertexA fan
A divider
G
P
GD 2011September 21,
2011
Future Works
Finding an algorithm that takes less than O(n4/3 + ɛ ) time.
Removing the constraint of the three outervertices from the NP-completeness result.
Work in progress:
Examining the time complexity of the point-set embedding problem in R2 when the input graph is 3-connected.
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GD 2011September 21,
2011
Thank You..
Dept. of Computer Science and Engg.Bangladesh University of
Engineering and Technology
Department of Computer ScienceUniversity of Victoria
Department of Computer ScienceUniversity of Manitoba