GD 2011September 21, 2011 Stephane Durocher Debajyoti Mondal Md. Saidur Rahman Sue Whitesides Rahnuma Islam Nishat Dept. of Computer Science and Engg

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


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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Point-Set Embeddings

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A plane graph G An embedding of G on P

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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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Plane 3-Trees

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A plane 3-tree G

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A construction for G

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a

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A plane 3-tree G

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The representative vertex of G

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A plane 3-tree

A plane 3-tree

A construction for G

Properties of Plane 3-trees

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A plane 3-tree

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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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A plane 3-tree G A point set P

We can map the outervertices in Six different ways.

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a a

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Find a valid mapping for the representative vertex.

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Valid mapping??

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n1 = 1

n2 = 1

n3 = 2

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Find a valid mapping for the representative vertex.


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fgh

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n1 = 1

n2 = 1

n3 = 2

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a a

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Find a valid mapping for the representative vertex recursively.

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How fast can we find a valid mapping for the representative vertex,

if such a mapping exists?

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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.

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fgh

Assume that n1≤ min{n2,n3}.

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n3 = 2

n2 = 1

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n1 = 1

n2 = 1

n3 = 2

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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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fgh

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Choose a random point z ϵΔabcWe need n3 points in this regionn1 = 1

n2 = 1

n3 = 2

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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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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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n1 = 1

n2 = 1

n3 = 2


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Extension to R3 ?

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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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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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3-Partition PSE in R3

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a 3mBB

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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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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

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a1

a2

a|S| A fan

A divider

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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:

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{10, 10, 12} { 9, 11, 12} { 9, 9,14} { 10, 11,11}


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a 1a 2

a 3mBB

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ca21

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BB

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A fan

A dividera

A spine vertex

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BB

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A fan

A dividera

A spine vertex

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BB

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EdgeCrossings?

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A spine vertexA fan

A divider

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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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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

Documents

GD 2011September 21, 2011 Stephane Durocher Debajyoti Mondal Md. Saidur Rahman Sue Whitesides Rahnuma Islam Nishat Dept. of Computer Science and Engg