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applications in which images have to be processed and some regions have to be searched for

and identified. When this processing is to be performed by a computer automatically without

the assistance of a human expert, a useful way of representing the knowledge is by using

attributed graphs. Attributed graphs have been proved as an effective way of representing

objects. When using graphs to represent objects or images, vertices usually represent regions

(or features) of the object or images, and edges between them represent the relations

between regions. Nonetheless planar graphs are graphs which can be drawn in the plane

without intersecting any edge between them. Most applications use planar graphs to

represent an image.

Graph matching (with attributes or not) represents an NP-complete problem, nevertheless

when we use planar graphs without attributes we can solve this problem in polynomial time

[1]. No algorithms have been presented that solve the attributed graph-matching problem and

use the planar-graphs properties. In this master thesis, we research about Attributed-Planar-

Graph matching. The aim is to find a fast algorithm through studying in depth the properties

and restrictions imposed by planar graphs.

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- 1. MEIS: ITINERARI SEGURETAT Raül Arlàndez Reverté Course 2009/2010
- 2. CONTENTS1. Objectives2. Definitions3. Planar Graph matching without attributes4. Planar Graph matching with attributes5. Practical evaluation6. Conclusion
- 3. 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.
- 4. CONTENTS1. Objectives2. Definitions 1. Planar Graph 2. Attributed planar Graph 3. Attributed Graph Matching 4. Tree decomposition3. Planar Graph matching without attributes4. Planar Graph matching with attributes5. Practical evaluation6. Conclusion
- 5. 2. DEFINITIONS1. Planar GraphGraph which can be drawn in the plane without intersecting any edgebetween them.Kuratowski’s theorem says:Graph is planar is not contain subgraph K5 or k3,3Also, 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
- 6. 2. DEFINITIONS 2. Attributed planar graph 3. Atrributed planar graph matching Object= window Colour= yellowObject= windowColour= white NP problem using restrictions(attributes and planar graphs ) can solve in polynomial time
- 7. 2. DEFINITIONS4. Tree decompositionAnother way to represent a planar graph W=2Tree width the size of the largest set X minus one
- 8. CONTENTS1. Objectives2. Definitions3. Planar Graph matching without attributes - Eppstein Algorithm - 1. Tree decomposition creation - 2. Generation of vertices combination - 3. Isomorphism list - 4. Sample Eppstein algorithm4. Planar Graph with attributes5. Practical evaluation6. Conclusion
- 9. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTESEppstein 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 creation2. Generation of vertices combinations (L(N), x and y) – Apply consistency conditions – edge function3. List and count how many isomorphism there are in the planar graph.
- 10. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES 1. Tree decomposition 2. Generation of combinations Subgraph L(N’) Subgraph L(N)Tree decomposition T of Graph G A { A, B, H, G }{ A, D, F, G } { A, B,C,D} L(N) D B{A, D, E, F } {A, B, C, G} x L(N’) G y Graph G Graph H Adding vertex x and y - x= L(N)-L(N’) = G - y= vertices not treated yet
- 11. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES2. Generation of combinations L(N): A B C D x={G} yL(N) Combinations x and y combinations General equation:Adding case:General equation:
- 12. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES2. Generation of combinationsFinal CombinationsIniatially, b=0
- 13. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES2. 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.
- 14. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES2. Generation of combinations• Edge function3. Isomorphism list
- 15. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES4. Sample: Eppstein Algorithm Partial isomorph boundary Induced Subgraph G1’Graph G1 Graph H
- 16. CONTENTS1. Objectives2. Definitions3. Planar Graph matching without attributes4. Planar Graph matching with attributes -Introduction -Attributed edge function -Euclidian distance -Sample 1 -Sample 25. Practical evaluation6. Conclusion
- 17. 4. PLANAR GRAPH MATCHING WITH ATTRIBUTES• There is no optimal solutions, at the moment• We propose new solution based on Eppstein’s Algorithm Object= window Colour= yellow Object= window Colour= white• Applications to Computer Vision• Attributes produce consistency reduction
- 18. 4. PLANAR GRAPH MATCHING WITH ATTRIBUTESNew Algorithm (Arlandez’s algorithm)1. Tree decomposition creation2. 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
- 19. 4. PLANAR GRAPH MATCHING WITH ATTRIBUTESNew Algorithm (Arlandez’s algorithm)• Attribute edge function • Tolerance variable that accepts values. • Two ways: • Exact method threshold=0 • Approximation method threshold >0
- 20. 4. PLANAR GRAPH MATCHING WITH ATTRIBUTESNew Algorithm (Arlandez’s algorithm) Graph H• Euclidian distance 1 Planar graph 2 2 1 3 # A D F G E x y C B H Euclidian distance 1 [] a [] d b [] [] c [] [] 0 2 [] a c d [] [] [] [] b [] 0 3 [] d [] a b [] [] c [] [] 0 4 [] d c a [] [] [] [] b [] 0 5 [] a [] d [] [] [] c b [] 10 6 [] a c d b [] [] [] [] [] 10 7 [] d [] a [] [] [] c b [] 10 8 [] d c a b [] [] [] [] [] 10
- 21. 4. PLANAR GRAPH MATCHING WITH ATTRIBUTESAttributed planar graph matching Partial isomorphism boundary• Sample 1 Induced Subgraph G1’Graph G1 Graph H 7 4 4 7 4 4 3 8 7 3 1 1 4
- 22. 4. PLANAR GRAPH MATCHING WITH ATTRIBUTES Attributed planar graph matching Partial isomorphism boundary • Sample 2 Induced Subgraph G1’ Graph HGraph G1 7 4 4 7 4 4 3 8 7 3 6 6 4 Threshold =1
- 23. CONTENTS1. Objectives2. Definitions3. Planar Graph matching without attributes4. Planar Graph matching with attributes5. Practical evaluation 1. Analysis: Eppstein VS Arlandez 2. Tree decomposition analysis 3. Atrributed approximation analysis6. Conclusion
- 24. 5. PRACTICAL EVALUATIONIt has used an application to do several tests
- 25. 5. PRACTICAL EVALUATIONPlanars Graphs G and Tree Decompositions Planar Graph G1 { A, F, G, H } { A, F, H, M } { A, E, F, M } { F, H, K, M } W=3 { A, D, E, M } { H, K, L, M } { A, C, D, E } { A, B, C, E } { A, B, H, G } Planar Graph G2 { A, B, D, G } { A, D, F, G } W=4 {A, B, C, D } {A, D, E, F }
- 26. 5. PRACTICAL EVALUATIONMore Tree Decompositions Planar Graph G1 { A, F, G, H } { A, F, H, M } { A, F, G, H, M }{ A, E, F, M } W=3 { F, H, K, M } { A, D, E, F, M } { F, H, K, L, M } W=4{ A, D, E, M } { H, K, L, M } {A, B, C, D, E }{ A, C, D, E } { A, B, C, E }
- 27. 5. PRACTICAL EVALUATIONMore Tree Decompositions Planar Graph G2 { A, D, F, G, E } { A, B, H, G } W=4 W=3 { A, B, G, H, D}{ A, B, D, G } { A, D, F, G }{A, B, C, D } {A, D, E, F } { A, B, D, G, C}
- 28. 5. PRACTICAL EVALUATION1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm possible partial combinations partial Test 1 isomorphism after applying isomorphism with combinations consistency edge relation hold Without With Without With Without With attributes attributes attributes attributes attributes attributes Step 1 join {ACDE} and {ABCE}Graph G1 3526 810 95 34 64 24 Step 2 join {ADEM} and {ACDE+B}W=3 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} andGraph H {AFHM+BCDLEK} 12096 2688 312 221 139 16 Partial Isomorphism {AFGHBCDLEKM} 60 6
- 29. 5. PRACTICAL EVALUATION1.Analysis: Eppstein’s Algorithm vs Arlandez’s AlgorithmTest 1 Combinations after applying Possible Partial isomorphism consistency combinations 80018000 70016000 6001400012000 50010000 400 8000 300 6000 200 4000 100 2000 0 0 Step 1 Step 2 Step 1 Step 2 Step 3 Step 3 Step 4 Step 4 Step 5 Step 5 Attributed Planar Graphs Step 6 Attributed Planar Step 6 Graph Planar graph without Planar Graph attributes without attributes
- 30. 5. PRACTICAL EVALUATION1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm Test 2 possible partial vertices=4 combinations after partial isomorphism with isomorphism Attributes= combinations applying consistency edge relation holdGraph G2 (a=5, b=5, c=2, d=1) Without With Without With Without With attributes attributes attributes attributes attributes attributesW=4 STEP 1 {A,B,G,H,D} join 168961 12502 663 149 437 115 with {A,B,D,G,C} STEP2 {A,D,F,G,E} join 293227 17365 3335 486 586 188 with { A,B,D,G,H+C,B} TOTAL 462188 29867 3998 635 1023 303 Partial Isomorphisms 169 72 Graph H
- 31. 5. PRACTICAL EVALUATION1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm Test 2: Possible partial isomorphism Combinations after applying combinations 293227 consistency 300000 3335 3500 250000 3000 200000 168961 2500 150000 2000 1500 100000 1000 663 50000 16524 706 32825 500 135 0 0 STEP 1 STEP2 STEP 1 STEP2 planar graph 2 with attributes… planar graph 2 with attributes (a=5, b=5,c=2, d=1) planar graph 2 without attributes planar graph 2 without attributes
- 32. 5. PRACTICAL EVALUATION1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm Test 2: Total combinations of planar graph 2 Total combinations of planar graph 2 matching with a square (vertex=4) matching a triangle (vertex=4) 29867 462188 500000 450000 30000 400000 25000 350000 300000 20000 250000 15000 200000 150000 10000 3273 49349 3998 841 100000 5000 635 184 50000 0 0 possible partial combinations after possible partial combinations after isomorphism applying consistency isomorphism applying consistency combinations combinations Planar graph without attributes (w=4) planar graph 2 without attributes w=4 Planar graph 2 with attributes (a=1, b=2, c=3 ) planar graph 2 with attributes (a=5, b=5, c=2, d=1)
- 33. 5. PRACTICAL EVALUATION 2. Tree decomposition analysis possible partial combinations after partial isomorphism with W=3 isomorphism applying consistency edge relation hold combinations STEP 1 {A,B,D,G} join 6862 107 94 Graph G2 Graph H with {A,B,C,D} STEP2 {A,D,F,G} join with 3268 74 64 { A,D,E,F} { A, F, G, H, M } STEP 3 join {A,B,H,G} between {A,B,D,G+C} and 17222 892 102{ A, D, E, F, M } { F, H, K, L, M } {A,D,F,G+E} TOTAL 27352 1073 260{A, B, C, D, E } Partial isomorphism 72 possible partial combinations after partial isomorphism with W=4 isomorphism applying consistency edge relation hold combinations STEP 1 {A,B,G,H,D} join { A, D, F, G, E } 12502 149 115 with {A,B,D,G,C} STEP2 {A,D,F,G,E} join 17365 486 188 with { A,B,D,G,H+C,B} { A, B, G, H, D} TOTAL 29867 635 295 Partial isomorphism 72 { A, B, D, G, C}
- 34. 5. PRACTICAL EVALUATION2. Tree decomposition analysis Total combinations before and after using consistency 27352 29867 30000 25000 W=3 W=4 20000 15000 10000 1073 635 5000 0 possible partial isomorphism combinations after applying combinations consistency
- 35. 5. PRACTICAL EVALUATION3.Attributed approximation analysis Graph H Possible combinations, threshold =4 3 Possible combinations, threshold =2 60000 60000 50000 7 9 40000 40000 30000 20000 20000 Graph G1 0 10000 0 planar graph 1 with attributes planar graph 1 without attributes planar graph 1 with attributes planar graph 1 without attributes
- 36. 5. PRACTICAL EVALUATION3.Attributed approximation analysis Graph H Possible Possible combinations, threshold 3 60000 combinations, threshold =7 60000 2,4,7 50000 50000 7 9 40000 40000 30000 Graph G1 30000 20000 20000 10000 10000 0 STEP 1 STEP 2 0 STEP 3 STEP 4 STEP 5 STEP 6 TOTAL STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 STEP 6 TOTAL planar graph 1 with attributes planar graph 1 with attributes (threshold=2) planar graph 1 wit attributes (threshold=4) planar graph 1 without attributes planar graph 1 with attributes (threshold=7) planar graph 1 without attributes
- 37. CONTENTS1. Objectives2. Definitions3. Planar Graph matching without attributes4. Planar Graph matching with attributes5. Practical evaluation6. Conclusion
- 38. 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
- 39. THANK YOU FOR YOUR ATTENTION

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