Rahnuma Islam Nishat Debajyoti Mondal Md. Saidur Rahman Graph Drawing and Information Visualization...

Rahnuma Islam NishatDebajyoti Mondal

Md. Saidur Rahman

Graph Drawing and Information Visualization LaboratoryDepartment of Computer Science and Engineering

Bangladesh University of Engineering and Technology (BUET)Dhaka-1000, Bangladesh

Point-Set Embeddings of Plane 3-Trees

a

b i

hf

e

d

cg

g

a

c

b

ih

ef

d

Point-Set Embedding

A Plane Graph G A Point-Set SPoint-Set Embedding of G

GD 2010, Konstanz, Germany September 24, 2010

Vertices are mapped to points, edges are drawn as straight-line segments

a

b

c

d

e

f

Point-Set Embedding

A triangle with 2 points

No triangle with 2 points

a

b cd

a

b c

d

d

a

b c

There exists no embedding of this graph on this point set


Previous Results

Graph Class Results

Bose et al. [1995] Trees

Gritzmann et. al. [1991] Outerplanar Graphs O(n 2)

O(n log n)

P. Bose [2002] Outerplanar Graphs O(n log3 n)

S. Cabello [2006] General Planar Graphs,2-Outerplanar Graphs

NP-Complete

Garcia et al. [2009] 3-connected cubic plane graphs

Ikebe et al. [1994]

Necessary and Sufficient Condition

Trees O(n2)


Our Results

Problems Results

Plane 3-Tree of n vertices,

point set of n points O(n2 log n)

Lower bound O(n log n)

Plane 3-Tree of n vertices,

point set of k points where k > n. nk8


a

b

c

d

e

f

A triangulated plane graph

Plane 3-Trees

a

b

c

Base case : a triangle

d

a

b

c

Insert vertex d

a

b

c

d

e

Insert vertex e

Insert vertex f


Representative Vertex of a Plane 3-TreePlane 3-Tree Plane 3-Tree

Plane 3-Tree

Representative Vertex

a

b

c

d

e

f

In a Plane 3-Tree there is an unique inner vertex which is the neighbor of all the three outer vertices

b

ce

d

a

d

b f

d

c

a


Outline of Algorithma

b

c

d

e

If G has a point-set embedding on S thenthe convex hull of S has exactly 3 points of S

Convex Hull

fgh

A Plane 3-Tree G A Point-Set S


Outline of Algorithm

b

c

d

e


a

fgh

c

a

b



b

c

d

e

a c

b

a

3C2 = 6 ways to map the outer vertices of G

to the 3 points on the convex hull of S

fgh



d


b

c

d

e

a a

c

b

n1=1

n2=1

n3=2 0

3

1

The number of points in each of the three regionscan be counted in O(n) time.

fgh

Map the Representative Vertex Valid mapping??



b

c

d

e

a a

c

bd

1

1

2

The mapping of the representative vertex is unique Finding a valid mapping of the representative vertex takes O(n2) time.

fgh

Map the Representative Vertex

n1=1

n2=1

n3=2



b

c

d

e

a a

c

bd

fgh

e

hf

g

The mapping of the other vertices are obtained recursively The algorithm computes the embedding of G in O(n3) time.

Map the other vertices Recursively



b

c

d

e

a a

c

bd

fgh

e

hf

g

The mapping of the other vertices are obtained recursively The algorithm computes the embedding of G in O(n3) time.

Map the other vertices Recursively

How to reduce the time complexity ??


Mapping of the representative vertex takes O(n2) time

O(n log n) time

O(n3) O(n2 log n)


Our Idea

We formulate a set of linear equations to obtain the unique mapping of the representative vertex

in O(n log n) time


Our Idea

u1

a

b

u2

u3

u4

u5

u1

u2

u5

u3u4

u5 u2 u4 u3 u1

1 2 3 4 5Ab

c

Sort the points according to their slopes

We use this array Ab to find the number of points above or

below the slope of a point


Our Idea


u1

a

c

b

u2

u3

u4

u5

u1u2 u3

u4,u5

u4 u5 u2 u1 u3

1(1) 2 3 4 5Aa

We use this array Aa to find the number of points on the left or on the right of the slope of a point


Our Idea

u1

a

bu2

u3

u4

u5

u1

u5

u3

u2,u4u3 u1 u2 u4 u5

1 2 3(1) 4 5Ac

c


We use this array Ac to find the number of points above or

below the slope of a point


Counting Number of Points

u1

a

bu2

u3

u4

u5

c



u1

a

bu2

u3

u4

u5

c

The interior of the convex hull is divided into 9 disjoint regions

Identify the disjoint regions

x5

x3x4

x6

x2

x9

x7

x8

x1



x5

a

b

x3x4

x6

x2

c

x1+x7+x6 = n1

x2+x8+x3 = n2

x4+x9+x5 = n3

x9

x7

x8

x1 u2 = db

c

d

e

a

fgh

n1=1

n2=1

n3=2

Formulate 3 constraints



Formulate 9 linear equations

x5

a

x3x4

x6

x2

c

x9

x7

x8

x1 u2 = d

x2+x1+x7+x6 = 2x5+x9+x4+x3 = 2

x8 = 0

u4 u5 u2 u1 u3

1(1) 2 3 4 5Aa

b


u5 u2 u4 u3 u1

1 2 3 4 5Ab

Set of Linear Equations


x5

a

x3x4

x6

x2

c

x9

x7

x8

x1 u2 = d

x2+x8+x3+x4 = 1x1+x7+x6+x5 = 3

x9 = 0

b


Ac



x5

a

x3x4

x6

x2

c

x9

x7

x8

x1 u2 = d

x6+x5+x9+x4 = 2x1+x2+x8+x3 = 1

x7 = 1

b

u3 u1 u2 u4 u5

1 2 3(1) 4 5



All the linear equations are given belowLinear Equations :

x2+x1+x7+x6 = 2x2+x8+x3+x4 = 1 x6+x5+x9+x4

= 2x5+x9+x4+x3 = 2x1+x7+x6+x5 = 3 x1+x2+x8+x3

= 1x8 = 0 x9 = 0 x7 =

1

Constraints :x1+x7+x6 = n1

x2+x8+x3 = n2

x4+x9+x5 = n3Solving these linear equations we find whether

this is a valid mapping in O(1) time


Complexity Analysis

Given the slope lists,a representative vertex can be found in O(n) time

Slope lists can be constructed for each representative vertex in O(n log n) time, which we use recursively to find

subsequent representative vertices

Complexity at each step is O(n log n+ n) =O(n log n)

Total complexity for n steps is = O(n) x O(n log n) =O(n2 log n)

This time complexity can be further improved to O(n2) if the points are in general position


Lower Bound

SORTING: A list of unsorted integers X=x1, x2, x3, … , xn, |X| = n

A set of n+2 points

S = (x1, x12), (x2, x2

2), (x3, x32),…, (xn, xn

2), (0,0), (xmax,0)

POINT-SET EMBEDDING

A plane 3-tree G with exactly one inner vertex of degree threesuch that deletion of the outer vertices gives a path of n-1 vertices


Lower Bound : Example

A set of 4+2=6 points

S = (5, 52), (2, 22), (3, 32),(9, 92), (0,0), (xmax=9,0)

Let the list of unsorted integers be 5, 2, 3, 9 where n = 4

(2, 22)

(3, 32)

(5, 52)

(0,0) (9,0)

(9, 92)



G

Exactly one inner vertex of degree three

a

b

c

d

ef

S

(2, 22)

(3, 32)

(5, 52)

(0,0) (9,0)

(9, 92)

A plane 3-tree G with exactly one inner vertex of degree three



A path of n-1=3 vertices

a c

d

ef

b

G S

(2, 22)

(3, 32)

(5, 52)

(0,0) (9,0)

(9, 92)

A plane 3-tree G with exactly one inner vertex of degree three

Deletion of the outer vertices gives a path of n-1 = 3 vertices



G a

b

c

d

ef

S

(2, 22)

(3, 32)

(5, 52)

(0,0) (9,0)

(9, 92)

We can show that from the point-set embedding of G we can find a sorted order of the points

G always has an embedding on S



G a

b

c

d

ef

S

(2, 22)

(3, 32)

(5, 52)

(0,0) (9,0)

(9, 92)

convex-hull of S contains (xmax=9, 92), (xmax=9,0), (0,0)

Find the convex hull of S


ca

b


a

b

c

d

ef

We show two mapping of the outer vertices a, b, c

Map the outer vertices of G

a

b

c


ca

b


a

b

c

d

ef

Map the representative vertex d

a

b

ca

ab

bc

c

0

0

2

dd


ca

b


a

b

c

d

ef

Map the representative vertex d

a

b

ca

ab

bc

c

0

0

2

d

d

Representative vertex d can be mapped to the point with second smallest x-coordinate or second largest x-coordinate


ca

b


a

b

c

d

ef

Map the other vertices recursively

a

b

ca

ab

bc

c

d

d

ee

f

f


ca

b


a

b

c

d

ef

a

b

ca

ab

bc

c

d

d

ee

f

f

The order in which the inner vertices are mappedgives the sorted order of the numbers

ascending descending


Lower Bound

Since the lower bound of Sorting Problem is Ω(n log n)

The Lower Bound of Point-Set Embedding = Ω(n log n)


What if |S| > n ?

b

c

d

e

a

fgh


n = 8 |S| = k = 20

The number of points in S is greater than the number of vertices of GGD 2010, Konstanz, Germany September 24, 2010

Representative Tree

b

c

a

A Plane 3-Tree G

If n=3 in G then T is a null tree

Representative Tree T

Base Case : n=3


Representative Tree

b

a

A Plane 3-Tree G

If n=4 in G then T has only one node with three dummy vertices

Representative Tree T

n=4

d

c

d

b

c

a


b

a

d

c

b d

e

Representative Tree

a

A Plane 3-Tree G Representative Tree T

Add other nodes to T recursively

c

d

e


b d

e

a

c

b d

e

Representative Tree

a



c

d

e

ff


d

e

a

c

f

b d

e

Representative Tree

a



c

d

e

fg

fg


b d

e

a

c

fg

b d

e

Representative Tree

a



c

d

e

fg

fg

h h

T is an unique ternary tree whose internal nodes are the n-3 inner vertices of G and leaves are dummy nodes


Our Algorithm

b

c

d

e

a

fgh


Take projection of the points of S on X and Y axis


Our Algorithm

b

c

d

e

a

fgh


Map the outer vertices to any 3 points of S

a

c

b


Our Algorithm

Representative tree T of G A Point-Set S

Map the root d of T to a point inside abca

c

bd(5)

e(1)

f(2)g(1)

h(1)

d


Subproblems

Representative tree T of G A Point-Set S

Map the root d of T to a point inside abca

c

bd(5)

e(1)

f(2)g(1)

h(1)

d

The mapping of d along with the outer vertices divide the problem into three subproblems


RecursionSolve the subproblems recursively for each mapping of d

a

c

b df(2)

g(1)

a

c

d



a

c

b df(2)

g(1)

a

c

d

f

No point for g



a

c

b df(2)

g(1)

a

c

dfalse



a

c

b da

c

d

a

b db d

cfalse



a

c

b da

c

d

a

b d

b d

c

d(5)

e(1)

f(2)g(1)

h(1)

false



a

c

d

a

b d

b d

c

d(5)

e(1)

f(2)g(1)

h(1)h

e

false


Combine the SolutionsCombine the solutions of the subproblems

a

c

d

a

b d

b d

c

false

h

e

a

c

b d

h

e

false


Overlapping Subproblems

a

c

b d

h

e

false

Store the results of the Subproblemsa

c

b

d

h

e

false

The results of the subproblems can be reused in overlapping cases


Complexity Analysis

Total time to compute a point set embedding (if one exists)is O(nk8)


Thank You

Rahnuma Islam Nishat Debajyoti Mondal Md. Saidur Rahman Graph Drawing and Information Visualization...

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Transcript of Rahnuma Islam Nishat Debajyoti Mondal Md. Saidur Rahman Graph Drawing and Information Visualization...