Attributed Graph Matching of Planar Graphs

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Many fields such as computer vision, scene analysis, chemistry and molecular biology haveapplications in which images have to be processed and some regions have to be searched forand identified. When this processing is to be performed by a computer automatically withoutthe assistance of a human expert, a useful way of representing the knowledge is by usingattributed graphs. Attributed graphs have been proved as an effective way of representingobjects. 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 relationsbetween regions. Nonetheless planar graphs are graphs which can be drawn in the planewithout intersecting any edge between them. Most applications use planar graphs torepresent an image. Graph matching (with attributes or not) represents an NP-complete problem, neverthelesswhen 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 anduse 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 propertiesand restrictions imposed by planar graphs.

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  • 1. MEIS: ITINERARI SEGURETATRal Arlndez RevertCourse 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.Kuratowskis 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= 3 and there are no cycles of length 3, then a=0 20. 4. PLANAR GRAPH MATCHING WITH ATTRIBUTESNew Algorithm (Arlandezs algorithm) Graph H Euclidian distance 1 Planar graph 22 13# ADFGExyCBHEuclidian distance1 [] a[] db[] []c [] []02 [] acd[] [] [] [] b[]03 [] d[] ab[] []c [] []04 [] dca[] [] [] [] b[]05 [] a[] d[] [] []c b[] 106 [] acdb[] [] [] [] [] 107 [] d[] a[] [] []c b[] 108 [] dcab[] [] [] [] [] 10 21. 4. PLANAR GRAPH MATCHING WITHATTRIBUTESAttributed planar graph matching Partial isomorphismboundary Sample 1Induced Subgraph G1Graph G1 Graph H74 474 4 3 8 73 1 1 4 22. 4. PLANAR GRAPH MATCHING WITHATTRIBUTES Attributed planar graph matchingPartial isomorphismboundary Sample 2 Induced Subgraph G1 Graph HGraph G1 7 4 474 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 DecompositionsPlanar 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 DecompositionsPlanar 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 DecompositionsPlanar Graph G2{ A, D, F, G, E }{ A, B, H, G }W=4W=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: Eppsteins Algorithm vs Arlandezs Algorithmpossible partialcombinations partialTest 1 isomorphismafter applying isomorphism with combinationsconsistency edge relation hold WithoutWith WithoutWith WithoutWithattributes attributes attributes attributes attributes attributes Step 1 join {ACDE} and {ABCE}Graph G13526 810 95 34 64 24 Step 2 join {ADEM} and {ACDE+B}W=3 52481248 15955 73 42 Step 3 join {AEFM} and {ADEM+BC} 6862 2268 215559725Step 4 join {FHKM} and {HKLM}6160 986 12536 85 22Step 5 join {AFHM} between{AEFM+BCD} and {FHKM+L}165622773 775 192108 20 Step 6 join {AFGH} andGraph H {AFHM+BCDLEK}120962688312221139 16Partial Isomorphism{AFGHBCDLEKM} 60 6 29. 5. PRACTICAL EVALUATION1.Analysis: Eppsteins Algorithm vs Arlandezs AlgorithmTest 1Combinations after applying Possible Partial isomorphismconsistency combinations 80018000 700160006001400012000 50010000 400 8000 3006000200400010020000 0Step 1 Step 2 Step 1 Step 2Step 3Step 3 Step 4Step 4Step 5Step 5Attributed Planar Graphs Step 6 Attributed PlanarStep 6GraphPlanar graph withoutPlanar Graphattributeswithout attributes 30. 5. PRACTICAL EVALUATION1.Analysis: Eppsteins Algorithm vs Arlandezs Algorithm Test 2 possible partial vertices=4 combinations after partial isomorphism withisomorphism Attributes=combinations applying consistency edge relation holdGraph G2 (a=5, b=5, c=2, d=1) Without With WithoutWithWithoutWith attributes attributes attributes attributesattributes attributesW=4STEP 1 {A,B,G,H,D} join 16896112502 663149 437115with {A,B,D,G,C} STEP2 {A,D,F,G,E} join 29322717365 3335 486 586188 with { A,B,D,G,H+C,B} TOTAL 46218829867 3998 635 1023 303Partial Isomorphisms169 72 Graph H 31. 5. PRACTICAL EVALUATION1.Analysis: Eppsteins Algorithm vs Arlandezs Algorithm Test 2: Possible partial isomorphismCombinations after applying combinations 293227consistency3000003335350025000030002000001689612500150000200015001000001000663 50000 1652470632825500 135 0 0 STEP 1STEP2STEP 1STEP2planar 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: Eppsteins Algorithm vs Arlandezs 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 45000030000 40000025000 350000 30000020000 25000015000 200000 15000010000 327349349 3998 841 100000 5000 635 1845000000 possible partialcombinations afterpossible partial combinations afterisomorphism applying consistency isomorphismapplying consistencycombinations 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 EVALUATION2. Tree decomposition analysispossible partialcombinations afterpartial isomorphism withW=3isomorphism applying consistencyedge relation hold combinationsSTEP 1 {A,B,D,G} join 686210794Graph G2Graph H with {A,B,C,D}STEP2 {A,D,F,G} join with 326874 64 { A,D,E,F}{ A, F, G, H, M }STEP 3 join {A,B,H,G}between {A,B,D,G+C} and 17222892102{ A, D, E, F, M } { F, H, K, L, M }{A,D,F,G+E} TOTAL27352 1073260{A, B, C, D, E } Partial isomorphism 72possible partialcombinations afterpartial isomorphism withW=4isomorphism applying consistencyedge relation hold combinationsSTEP 1 {A,B,G,H,D} join{ A, D, F, G, E } 12502149115with {A,B,D,G,C} STEP2 {A,D,F,G,E} join17365486188 with { A,B,D,G,H+C,B}{ A, B, G, H, D}TOTAL 29867635295Partial isomorphism72{ A, B, D, G, C} 34. 5. PRACTICAL EVALUATION2. Tree decomposition analysisTotal combinations before and after using consistency27352 29867 30000 25000W=3 W=4 20000 15000 1000010736355000 0 possible partial isomorphism combinations after applying combinationsconsistency 35. 5. PRACTICAL EVALUATION3.Attributed approximation analysis Graph H Possible combinations, threshold =4 3 Possible combinations, threshold =2 60000 60000 50000 79 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 HPossible Possible combinations, threshold3 60000combinations, threshold =7 600002,4,7 50000 50000 79400004000030000 Graph G1 3000020000 2000010000 100000 STEP 1 STEP 2 0 STEP 3 STEP 4 STEP 5STEP 6 TOTALSTEP 1 STEP 2STEP 3 STEP 4 STEP 5 STEP 6 TOTALplanar 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 Eppsteins 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