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Attributed Graph Matching of Planar Graphs

Raül Arlàndez RevertéCourse 2009/2010

MEIS: ITINERARI SEGURETAT

CONTENTS

1. Objectives2. Definitions3. Planar Graph matching without attributes4. Planar Graph matching with attributes5. Practical evaluation6. Conclusion

1. OBJECTIVES

• Why Graph? Images can be represented by graphs

• Why attributed graph? More semantic information

• Why Planar attributed graph? reduce combinations

• Why attributed planar graph matching ? Currently there is no papers about it.

CONTENTS

1. Objectives2. Definitions

1. Planar Graph2. Attributed planar Graph3. Attributed Graph Matching4. Tree decomposition

3. Planar Graph matching without attributes4. Planar Graph matching with attributes5. Practical evaluation6. Conclusion

2. DEFINITIONS1. Planar GraphGraph which can be drawn in the plane without intersecting any edge between them.Kuratowski’s theorem says:Graph is planar is not contain subgraph K5 or k3,3

Also, there is an algorithm to determine whether a graph is planar• Theorem 1: If 1 n>= 3 then a=<3n-6 (where n is the number of vertices and a the

numbers of edges) • Theorem 2: If n> 3 and there are no cycles of length 3, then a=<2n -4

2. DEFINITIONS2. Attributed planar graph

3. Atrributed planar graph matching

NP problem using restrictions(attributes and planar graphs ) can solve in polynomial time

Object= windowColour= white

Object= windowColour= yellow

2. DEFINITIONS4. Tree decompositionAnother way to represent a planar graph

Tree width the size of the largest set X minus one

W=2

CONTENTS

1. Objectives2. Definitions3. Planar Graph matching without attributes

- Eppstein Algorithm- 1. Tree decomposition creation- 2. Generation of vertices combination- 3. Isomorphism list- 4. Sample Eppstein algorithm

4. Planar Graph with attributes5. Practical evaluation6. Conclusion

3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES

Eppstein proposes an algorithm matching between a planar graph G and a graph H.–Find how many isomorphism there are between them.

Algorithm:

1. Tree decomposition creation

2. Generation of vertices combinations (L(N), x and y)– Apply consistency conditions – edge function

3. List and count how many isomorphism there are in the planar graph.


1. Tree decomposition

{ A, B, H, G }

{ A, D, F, G } { A, B,C,D}

{A, D, E, F } {A, B, C, G}

2. Generation of combinations

Adding vertex x and y- x= L(N)-L(N’) = G- y= vertices not treated yet

Graph H

Tree decomposition T of Graph G

Graph G

L(N)

L(N’)

A

DB

G

Subgraph L(N’) Subgraph L(N)

x

y


2. Generation of combinationsL(N) Combinations

Adding case:

General equation:

x and y combinations

General equation:

A B C D x={G} yL(N):


2. Generation of combinationsFinal Combinations

Iniatially, b=0


2. Generation of combinations• Consistency Suppose that node N has children N1 and N2. We say that two partial

isomorphisms: B: H→G’N and B1: H → G’N1 or, B: H→G’N and B2: H → G’N2 are consistent if the following conditions all hold:

1. For each vertex v є H, if B(v) є L(N1) or B1(v) є L (N), then B(v)= B1(v)

2. For each vertex v є H, if B(v) ≠ X , then B1(v) є L(N)U {Y}

3. At least one vertex v є H has B1(v) є L(N) {Y1} or B2(v) є L(N) U {Y2} if we have B1 and B2, otherwise it does not apply the condition.

4. For each partial isomorphism it must hold that if B(v)= X and B1(v) =y1 a partial possible isomorphism is discarded.

5. For each v with B(v)=x, exactly one of B1(v) and one of B2(v) is equal to y.


2. Generation of combinations• Edge function

3. Isomorphism list


4. Sample: Eppstein AlgorithmPartial isomorph

boundaryGraph G1

Graph H

Induced Subgraph G1’

CONTENTS

1. Objectives2. Definitions3. Planar Graph matching without attributes4. Planar Graph matching with attributes

-Introduction -Attributed edge function-Euclidian distance-Sample 1-Sample 2

5. Practical evaluation6. Conclusion

4. PLANAR GRAPH MATCHING WITH ATTRIBUTES

• There is no optimal solutions, at the moment• We propose new solution based on Eppstein’s Algorithm

• Applications to Computer Vision• Attributes produce consistency reduction

Object= windowColour= white

Object= windowColour= yellow


New Algorithm (Arlandez’s algorithm)

1. Tree decomposition creation

2. Generation of vertices combinations • Apply consistency conditions • edge attribute function (exact or approximation method)

3. List and count how many isomorphism there are in the planar graph.

4. Compute the euclidian distance of the isomorphism list• Sorting the distances from lesser to greater value


New Algorithm (Arlandez’s algorithm)• Attribute edge function• Tolerance variable that accepts values.• Two ways:• Exact method threshold=0• Approximation method threshold >0


New Algorithm (Arlandez’s algorithm)• Euclidian distance

# A D F G E x y C B H Euclidian distance1 [] a [] d b [] [] c [] [] 02 [] a c d [] [] [] [] b [] 03 [] d [] a b [] [] c [] [] 04 [] d c a [] [] [] [] b [] 05 [] a [] d [] [] [] c b [] 106 [] a c d b [] [] [] [] [] 107 [] d [] a [] [] [] c b [] 108 [] d c a b [] [] [] [] [] 10

Planar graph 2

1

1

2

3

Graph H


Attributed planar graph matching • Sample 1

Partial isomorphism boundary

7

4

1

4

83

4

7

4

7

1

4

3

Induced Subgraph G1’Graph HGraph G1


Attributed planar graph matching • Sample 2

Partial isomorphism boundary

7

4

6

4

83

4

7

4

7

6

4

3

Threshold =1

Induced Subgraph G1’Graph H

Graph G1

CONTENTS

1. Objectives2. Definitions3. Planar Graph matching without attributes4. Planar Graph matching with attributes5. Practical evaluation

1. Analysis: Eppstein VS Arlandez2. Tree decomposition analysis3. Atrributed approximation analysis

6. Conclusion

5. PRACTICAL EVALUATION

It has used an application to do several tests


{ A, F, G, H }

{ A, F, H, M }

{ A, E, F, M } { F, H, K, M }

{ A, D, E, M } { H, K, L, M }

{ A, C, D, E }

{ A, B, C, E }

{ A, B, H, G }

{ A, B, D, G } { A, D, F, G }

{A, B, C, D } {A, D, E, F }

Planars Graphs G and Tree Decompositions

W=3

W=4

Planar Graph G1

Planar Graph G2


{ A, F, G, H }

{ A, F, H, M }

{ A, E, F, M } { F, H, K, M }

{ A, D, E, M } { H, K, L, M }

{ A, C, D, E }

{ A, B, C, E }

More Tree Decompositions

{ A, F, G, H, M }

{ A, D, E, F, M } { F, H, K, L, M }

{A, B, C, D, E }

W=3W=4

Planar Graph G1


{ A, B, H, G }

{ A, B, D, G } { A, D, F, G }

{A, B, C, D } {A, D, E, F }

More Tree Decompositions

{ A, D, F, G, E }

{ A, B, G, H, D}

{ A, B, D, G, C}

W=3W=4

Planar Graph G2

5. PRACTICAL EVALUATION1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm

possible partial isomorphism combinations

combinations after applying

consistency

partial isomorphism with edge relation hold

Without attributes

With attributes

Without attributes

With attributes

Without attributes

With attributes

Step 1 join {ACDE} and {ABCE}3526 810 95 34 64 24

Step 2 join {ADEM} and {ACDE+B}5248 1248 159 55 73 42

Step 3 join {AEFM} and {ADEM+BC}6862 2268 215 55 97 25

Step 4 join {FHKM} and {HKLM} 6160 986 125 36 85 22

Step 5 join {AFHM} between {AEFM+BCD} and {FHKM+L} 16562 2773 775 192 108 20

Step 6 join {AFGH} and {AFHM+BCDLEK} 12096 2688 312 221 139 16

Partial Isomorphism {AFGHBCDLEKM} 60 6

Graph H

Graph G1

W=3

Test 1


Test 1

Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

0

2000

4000

6000

8000

10000

12000

14000

16000

18000Possible Partial isomorphism

combinations

Attributed Planar Graphs

Planar graph without attributes

Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

0

100

200

300

400

500

600

700

800Combinations after applying

consistency

Attributed Planar Graph Planar Graph without attributes


vertices=4Attributes= (a=5, b=5,

c=2, d=1)


combinations after applying consistency


Without attributes

With attributes

Without attributes

With attributes

Without attributes

With attributes

STEP 1 {A,B,G,H,D} join with {A,B,D,G,C} 168961 12502 663 149 437 115

STEP2 {A,D,F,G,E} join with { A,B,D,G,H+C,B} 293227 17365 3335 486 586 188

TOTAL 462188 29867 3998 635 1023 303Partial Isomorphisms 169 72

Test 2

Graph G2

W=4

Graph H

5. PRACTICAL EVALUATION1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm Test 2:

STEP 1 STEP2

0

50000

100000

150000

200000

250000

300000

1652432825

168961

293227

planar graph 2 with attributes (a=5, b=5,c=2, d=1)

STEP 1 STEP2

0

500

1000

1500

2000

2500

3000

3500

135706

663

3335

planar graph 2 with attributes (a=5, b=5,c=2, d=1)planar graph 2 without attributes

Possible partial isomorphism combinations

Combinations after applying consistency

5. PRACTICAL EVALUATION1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm Test 2:

possible partial isomorphism com-

binations


050000

100000150000200000250000300000350000400000450000500000

462188

399849349 841

planar graph 2 without attributes w=4

planar graph 2 with attributes (a=5, b=5, c=2, d=1)

possible partial isomorphism com-

binations


05000

1000015000200002500030000

29867

6353273

184

Planar graph without attributes (w=4)Planar graph 2 with attributes (a=1, b=2, c=3 )

Total combinations of planar graph 2 matching with a square (vertex=4)

Total combinations of planar graph 2 matching a triangle (vertex=4)

5. PRACTICAL EVALUATION2. Tree decomposition analysis

W=3possible partial

isomorphism combinations



STEP 1 {A,B,D,G} join with {A,B,C,D} 6862 107 94

STEP2 {A,D,F,G} join with { A,D,E,F} 3268 74 64

STEP 3 join {A,B,H,G} between {A,B,D,G+C} and

{A,D,F,G+E}17222 892 102

TOTAL 27352 1073 260Partial isomorphism 72

W=4possible partial

isomorphism combinations



STEP 1 {A,B,G,H,D} join with {A,B,D,G,C} 12502 149 115

STEP2 {A,D,F,G,E} join with { A,B,D,G,H+C,B} 17365 486 188

TOTAL 29867 635 295Partial isomorphism 72

Graph HGraph G2

{ A, F, G, H, M }

{ A, D, E, F, M } { F, H, K, L, M }

{A, B, C, D, E }

{ A, D, F, G, E }

{ A, B, G, H, D}

{ A, B, D, G, C}

5. PRACTICAL EVALUATION2. Tree decomposition analysis



0

5000

10000

15000

20000

25000

3000027352

1073

29867

635

W=3 W=4

Total combinations before and after using consistency

5. PRACTICAL EVALUATION3.Attributed approximation analysis

STEP

1

STEP

2

STEP

3

STEP

4

STEP

5

STEP

6

TOTA

L

0

10000

20000

30000

40000

50000

60000 Possible combinations, threshold =2

planar graph 1 with attributes

planar graph 1 without attributesSTEP 1

STEP 2

STEP 3

STEP 4

STEP 5

STEP 6

TOTAL

0

10000

20000

30000

40000

50000

60000Possible combinations, threshold =4

planar graph 1 with attributes

planar graph 1 without attributes

3

97

Graph H

Graph G1

5. PRACTICAL EVALUATION3.Attributed approximation analysis

STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 STEP 6 TOTAL

0

10000

20000

30000

40000

50000

60000

Possible combinations, threshold =7

planar graph 1 with attributesplanar graph 1 without attributes

STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 STEP 6 TOTAL

0

10000

20000

30000

40000

50000

60000Possible combinations, threshold

2,4,7

planar graph 1 with attributes (threshold=2)planar graph 1 wit attributes (threshold=4)planar graph 1 with attributes (threshold=7)planar graph 1 without attributes

3

97

Graph H

Graph G1

CONTENTS

1. Objectives2. Definitions3. Planar Graph matching without attributes4. Planar Graph matching with attributes5. Practical evaluation6. Conclusion

6. CONCLUSION

• Algorithm based on Eppstein’s algorithm.• Nowadays, writing to publish in a congress• Attributes reduce the number of combinations.• Edges function most important function.• Working with :

– high threshold worse – Low threshold better

• Optimal algorithms supposes great time expenditure• Euclidian distance make easier the best solution • Future work:

– Make our spanning tree given a planar graph– Work with no constant size tree decomposition

THANK YOU FOR YOUR ATTENTION