Katia Abbaci

1
A Similarity Skyline Approach for Handling
GraphQueries - A Preliminary Report

• Katia Abbaci Allel Hadjali Ludovic
  Liétard Daniel Rocacher
• IRISA/ENSSAT, University of Rennes1
• Katia.Abbaci, Allel.Hadjali,
  Daniel.Rocacher_at_enssat.fr
• IRISA/IUT, University of Rennes1
• Ludovic.Lietard_at_univ-rennes1.fr

2
Outline

• Introduction
• Background
• Skyline Query
• Graph Query
• Graph Similarity Measures
• Graph Similarity Skyline
• Refinement Graph Similarity Skyline
• Summary and Outlook

3
Introduction (1/3)

• Context
• Graphs Modeling of structured and complex data
• Application Domains
• Medicine, Web, Chemistry, Imaging, XML documents,
  Bioinformatic,...

GDM 2011
4
Introduction (2/3)

• Main
• Search Problem of similar graphs to graph query
• Existing approaches a single similarity measure
• Several methods for measuring the similarity betwe
  en two graphs
• Method limited to an application class
• No method fits all

5
Introduction (3/3)

• Motivations
• Model for different classes of applications
• Model incorporating multiple features
• Contributions
• Graph Similarity Skyline in order to answer a
  graph query optimality in the sense of Pareto
• A Refinement Method of Skyline based on diversity
  criterion among graphs

6
Skyline Query

• Identification of interesting objects from
  multi-dimensional dataset
• p (p1, , pm), q (q1, , qm)
  multidimensional objects
• p Pareto dominates q, denoted p q, iff
• on each dimension, 1 i m, pi qi
• on at least one dimension, pj lt qj

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Sample Skyline Query

• Find a cheap hotel and as close as possible
  to the downtown

H2
H2
H6
H6
Skyline H2, H4, H6
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Graph Query

• Two categories of graph queries
• Graph containment search
• q a query, D g1, , gn a GDB
• Subgraph containment search
• ? Retrieve all graphs gi of D such that q ? gi
• Supergraph containment search
• ? Retrieve all graphs gi of D such that q ? gi
• Graph similarity search
• Retrieve structurally similar graphs to the query
  graph

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Graph Similarity Measures

• Several processing methods of graph similarity
• Edit Distance (DistEd)
• Maximum common subgraph based distance (DistMcs)
• Graph union based distance (DistGu)

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Graph Similarity Measures
Distance between g and g Similarity between g and g
Edit Distance
Mcs-based Distance
Gu-based Distance
Tab. 2 Similarity Measures
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Edit Distance example

• Transformation of g into g
• deletion of the adge (d, e),
• re-labeling the adge (a, d) from 1 to 4,
• re-labeling the node d with e,
• insertion of the adge (a, f) with the label 1.
• Use of the uniform distance

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Distances based on Mcs and Gu example

• Identification of the size of
• Computation of Mcs-based distance
• Computation of Gu-based distance

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Graph Similarity Skyline (1/2)

• Graph compound similarity between two graphs a
  vector of local distance measures

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Graph Similarity Skyline (2/2)

• q a query, D g1, , gn a GDB
• For i 1 to n, do
• Compare
• Extract the Graph Similarity Skyline (GSS)
• Similarity-Dominance Relation
• ? i ? 1, ..., d, Disti(g, q) Disti(g, q),
• ? k ? 1, ..., d, Distk(g, q) lt Distk(g, q).

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Illustrative Example (1/2)
Mcs(gi, q)
(g1, q) 4
(g2, q) 4
(g3, q) 4
(g4, q) 3
(g5, q) 5
(g6, q) 5
(g7, q) 6
Tab. 3 Information about Mcs(gi, q)
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Illustrative Example (2/2)

• Computation of GCS(gi,q), for i 1 to 7, do

DistEd(gi,q) DistMcs(gi,q) DistGu(gi,q)
(g1, q) 4 0.33 0.50
(g2, q) 4 0.43 0.56
(g3, q) 3 0.43 0.56
(g4, q) 2 0.50 0.67
(g5, q) 3 0.38 0.44
(g6, q) 4 0.44 0.50
(g7, q) 4 0.40 0.40
g1
g5
g1
Tab. 4 Distance Measures
GSS(D, q) g1, g4, g5, g7
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Refinement of Graph Similarity Skyline (1/3)

• Large Skyline
• Need k dissimilar answers
• Solution diversity criterion
• Extract a subset (S) of size k with a maximal
  diversity
• Provide the user with a global picture of
  the whole set GSS

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Refinement of Graph Similarity Skyline (2/3)

• Diversity of a subset S of size k is
• diversity in the ith dimension of the subset
  S
• s. t.

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Refinement of Graph Similarity Skyline (3/3)

• Refinement Algorithm
• For j 1 to , enumerate
  , with
• For i 1 to d, rank-order all Sj in decreasing
  way according to their diversity
• Let be the rank of Sj w. r. t.
  the ith dimension
• the best diversity value
• the worst diversity value
• Evaluate Sj by
• Extract

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Illustrative Exa mple

• Return the 2 best graphs

v3
0.80
0.60
0.67
0.73
0.77
0.61

S1g1,g4
S2g1,g5
S3g1,g7
S4g4,g5
S5g4,g7
S6g5,g7
v1
0.86
0.83
0.87
0.80
0.83
0.75
r3
1
6
4
3
2
5

S1g1,g4
S2g1,g5
S3g1,g7
S4g4,g5
S5g4,g7
S6g5,g7
r1
2
3
1
4
3
5
v3
0.80
0.60
0.67
0.73
0.77
0.61
Val(Si)
5
14
9
10
6
15
v1
0.86
0.83
0.87
0.80
0.83
0.75
v2
0.67
0.50
0.60
0.62
0.70
0.50
v2
0.67
0.50
0.60
0.62
0.70
0.50
r2
2
5
4
3
1
5
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Summary and Outlook

• Skyline approach for searching graphs
  by similarity
• Extraction of all DB graphs non-dominated by
  any other graph
• Preserving information about the
  similarity on different features
• Selection of the subset of graphs with maximal
  diversity from the skyline
• Implementation step to demonstrate the
  effectiveness of the approach on a real database
• Investigation of other similarity measures

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Thank you
Questions ?