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MEIS: ITINERARI SEGURETAT




              Raül Arlàndez Reverté
              Course 2009/2010
CONTENTS

1.   Objectives
2.   Definitions
3.   Planar Graph matching without attributes
4.   Planar Graph matching with attributes
5.   Practical evaluation
6.   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. Objectives
2. Definitions
   1. Planar Graph
   2. Attributed planar Graph
   3. Attributed Graph Matching
   4. Tree decomposition
3.   Planar Graph matching without attributes
4.   Planar Graph matching with attributes
5.   Practical evaluation
6.   Conclusion
2. DEFINITIONS
1. Planar Graph
Graph 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. DEFINITIONS
    2. Attributed planar graph




    3. Atrributed planar graph matching
                                                  Object= window
                                                  Colour= yellow
Object= window
Colour= white




    NP problem using restrictions(attributes and planar
       graphs ) can solve in polynomial time
2. DEFINITIONS
4. Tree decomposition
Another way to represent a planar graph



                                          W=2




Tree width the size of the largest set X minus one
CONTENTS
1. Objectives
2. Definitions
3. 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 attributes
5. Practical evaluation
6. 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.
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
3. PLANAR GRAPH MATCHING
                   WITHOUT ATTRIBUTES
2. Generation of combinations   L(N):
                                        A   B   C     D   x={G}   y



L(N) Combinations                 x and y combinations




                                  General equation:


Adding case:
General equation:
3. PLANAR GRAPH MATCHING
                 WITHOUT ATTRIBUTES
2. Generation of combinations
Final Combinations




Iniatially, b=0
3. PLANAR GRAPH MATCHING
                   WITHOUT ATTRIBUTES
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.
3. PLANAR GRAPH MATCHING
                WITHOUT ATTRIBUTES
2. Generation of combinations
• Edge function




3. Isomorphism list
3. PLANAR GRAPH MATCHING
               WITHOUT ATTRIBUTES
4. Sample: Eppstein Algorithm    Partial isomorph
                                    boundary        Induced Subgraph G1’
Graph G1
                       Graph H
CONTENTS
1.   Objectives
2.   Definitions
3.   Planar Graph matching without attributes
4.   Planar Graph matching with attributes
     -Introduction
     -Attributed edge function
     -Euclidian distance
     -Sample 1
     -Sample 2
5. Practical evaluation
6. Conclusion
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
4. PLANAR GRAPH MATCHING WITH
                ATTRIBUTES
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
4. PLANAR GRAPH MATCHING WITH
               ATTRIBUTES
New Algorithm (Arlandez’s algorithm)
• Attribute edge function
  • Tolerance variable that accepts values.
  • Two ways:
     • Exact method  threshold=0
     • Approximation method  threshold >0
4. PLANAR GRAPH MATCHING WITH
               ATTRIBUTES
New 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
4. PLANAR GRAPH MATCHING WITH
                  ATTRIBUTES
Attributed 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
4. PLANAR GRAPH MATCHING WITH
                      ATTRIBUTES
 Attributed planar graph matching        Partial isomorphism
                                              boundary
 • Sample 2                                                 Induced Subgraph G1’
                               Graph H
Graph G1
           7                   4
                   4
                                                        7                  4
                           4
               3
   8                                 7                             3
                       6                                                   6

               4

                   Threshold =1
CONTENTS

1.   Objectives
2.   Definitions
3.   Planar Graph matching without attributes
4.   Planar Graph matching with attributes
5.   Practical evaluation
     1. Analysis: Eppstein VS Arlandez
     2. Tree decomposition analysis
     3. Atrributed approximation analysis
6. Conclusion
5. PRACTICAL EVALUATION
It has used an application to do several tests
5. PRACTICAL EVALUATION
Planars 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 }
5. PRACTICAL EVALUATION
More 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 }
5. PRACTICAL EVALUATION
More 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}
5. PRACTICAL EVALUATION
1.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} and
Graph H             {AFHM+BCDLEK}              12096        2688        312        221        139           16
                  Partial Isomorphism
                    {AFGHBCDLEKM}                                                               60           6
5. PRACTICAL EVALUATION
1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm
Test 1
                                                                          Combinations after applying
         Possible Partial isomorphism                                            consistency
                 combinations                                   800
18000                                                           700
16000
                                                                600
14000
12000                                                           500
10000                                                           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
5. PRACTICAL EVALUATION
1.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 hold
Graph G2       (a=5, b=5, c=2, d=1)
                                       Without       With       Without        With        Without        With
                                       attributes attributes   attributes   attributes    attributes   attributes
W=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
5. PRACTICAL EVALUATION
1.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
5. PRACTICAL EVALUATION
1.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)
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}
5. PRACTICAL EVALUATION
2. 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
5. PRACTICAL EVALUATION
3.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
5. PRACTICAL EVALUATION
3.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
CONTENTS

1. Objectives
2. Definitions
3.   Planar Graph matching without attributes
4.   Planar Graph matching with attributes
5.   Practical evaluation
6.   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

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Attributed Graph Matching of Planar Graphs

  • 1. MEIS: ITINERARI SEGURETAT Raül Arlàndez Reverté Course 2009/2010
  • 2. CONTENTS 1. Objectives 2. Definitions 3. Planar Graph matching without attributes 4. Planar Graph matching with attributes 5. Practical evaluation 6. 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. CONTENTS 1. Objectives 2. Definitions 1. Planar Graph 2. Attributed planar Graph 3. Attributed Graph Matching 4. Tree decomposition 3. Planar Graph matching without attributes 4. Planar Graph matching with attributes 5. Practical evaluation 6. Conclusion
  • 5. 2. DEFINITIONS 1. Planar Graph Graph 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
  • 6. 2. DEFINITIONS 2. Attributed planar graph 3. Atrributed planar graph matching Object= window Colour= yellow Object= window Colour= white NP problem using restrictions(attributes and planar graphs ) can solve in polynomial time
  • 7. 2. DEFINITIONS 4. Tree decomposition Another way to represent a planar graph W=2 Tree width the size of the largest set X minus one
  • 8. CONTENTS 1. Objectives 2. Definitions 3. 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 attributes 5. Practical evaluation 6. Conclusion
  • 9. 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.
  • 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 ATTRIBUTES 2. Generation of combinations L(N): A B C D x={G} y L(N) Combinations x and y combinations General equation: Adding case: General equation:
  • 12. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES 2. Generation of combinations Final Combinations Iniatially, b=0
  • 13. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES 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.
  • 14. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES 2. Generation of combinations • Edge function 3. Isomorphism list
  • 15. 3. PLANAR GRAPH MATCHING WITHOUT ATTRIBUTES 4. Sample: Eppstein Algorithm Partial isomorph boundary Induced Subgraph G1’ Graph G1 Graph H
  • 16. CONTENTS 1. Objectives 2. Definitions 3. Planar Graph matching without attributes 4. Planar Graph matching with attributes -Introduction -Attributed edge function -Euclidian distance -Sample 1 -Sample 2 5. Practical evaluation 6. 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 ATTRIBUTES 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
  • 19. 4. PLANAR GRAPH MATCHING WITH ATTRIBUTES New 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 ATTRIBUTES New 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 ATTRIBUTES Attributed 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 H Graph G1 7 4 4 7 4 4 3 8 7 3 6 6 4 Threshold =1
  • 23. CONTENTS 1. Objectives 2. Definitions 3. Planar Graph matching without attributes 4. Planar Graph matching with attributes 5. Practical evaluation 1. Analysis: Eppstein VS Arlandez 2. Tree decomposition analysis 3. Atrributed approximation analysis 6. Conclusion
  • 24. 5. PRACTICAL EVALUATION It has used an application to do several tests
  • 25. 5. PRACTICAL EVALUATION Planars 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 EVALUATION More 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 EVALUATION More 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 EVALUATION 1.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} and Graph H {AFHM+BCDLEK} 12096 2688 312 221 139 16 Partial Isomorphism {AFGHBCDLEKM} 60 6
  • 29. 5. PRACTICAL EVALUATION 1.Analysis: Eppstein’s Algorithm vs Arlandez’s Algorithm Test 1 Combinations after applying Possible Partial isomorphism consistency combinations 800 18000 700 16000 600 14000 12000 500 10000 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 EVALUATION 1.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 hold Graph G2 (a=5, b=5, c=2, d=1) Without With Without With Without With attributes attributes attributes attributes attributes attributes W=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 EVALUATION 1.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 EVALUATION 1.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 EVALUATION 2. 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 EVALUATION 3.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 EVALUATION 3.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. CONTENTS 1. Objectives 2. Definitions 3. Planar Graph matching without attributes 4. Planar Graph matching with attributes 5. Practical evaluation 6. 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