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A Similarity Skyline Approach for Handling Graph Queries - A Preliminary Report. Katia Abbaci† Allel Hadjali † Ludovic Liétard ‡ Daniel Rocacher † † IRISA/ENSSAT, University of Rennes1 {Katia.Abbaci , Allel.Hadjali, Daniel.Rocacher}@enssat.fr ‡ IRISA/IUT, University of Rennes1

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A similarity skyline approach for handling graph queries a preliminary report

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}@enssat.fr

‡IRISA/IUT, University of Rennes1

Ludovic.Lietard@univ-rennes1.fr


Outline
Outline

  • Introduction

  • Background:

    • Skyline Query

    • Graph Query

    • Graph Similarity Measures

  • Graph Similarity Skyline

  • Refinement Graph Similarity Skyline

  • Summary and Outlook

GDM 2011


Introduction 1 3
Introduction (1/3)

Context:

  • Graphs: Modeling of structured and complex data

  • Application Domains:

    • Medicine, Web, Chemistry, Imaging, XML documents, Bioinformatic,...

Chemistry

Web

Imaging

Medicine

GDM 2011


Introduction 2 3
Introduction (2/3)

Main:

  • Search Problem of similar graphs to graph query

    • Existing approaches: a single similarity measure

  • Several methods for measuring the similarity between two graphs:

    • Method limited to an application class

    • No method fits all

GDM 2011


Introduction 3 3
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

GDM 2011


Skyline query
SkylineQuery

  • Identification of interesting objects from multi-dimensional dataset

  • p = (p1, …, pm),q = (q1, …, qm): multidimensional objects

    p Pareto dominatesq, denoted pq, iff:

    • on each dimension, 1 ≤ i ≤ m, pi ≤ qi

    • on at least one dimension, pj < qj

GDM 2011


Sample skyline query
SampleSkylineQuery

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

H2

H2

H6

H6

Skyline = {H2, H4, H6}

GDM 2011

Tab. 1 –Sample of hotels


Graph query
Graph Query

  • Twocategories of graph queries:

    • Graph containmentsearch:

      q: a query, D = {g1, …, gn} a GDB

      • Subgraphcontainmentsearch

      •  Retrieve all graphs gi of D suchthatq ⊆ gi

      • Supergraphcontainmentsearch

         Retrieve all graphs gi of D suchthatq ⊇ gi

    • Graph similaritysearch:

      Retrieve structurally similar graphs to the query graph

GDM 2011


Graph similarity measures
Graph SimilarityMeasures

  • Several processing methods of graph similarity:

    • Edit Distance (DistEd)

    • Maximum common subgraph based distance (DistMcs)

    • Graph union based distance (DistGu)

GDM 2011


Graph similarity measures1
Graph SimilarityMeasures

Tab. 2 –SimilarityMeasures

GDM 2011


Edit distance example
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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Fig. 3 –Example of labeled graphs

Fig. 3 –Example of labeled graphs

Fig. 3 –Example of labeled graphs

Fig. 3 –Example of labeled graphs

Fig. 3 –Example of labeled graphs

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GDM 2011


Distances based on mcs and gu example
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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Fig. 4 –Example of labeled graphs

Fig. 4 –Example of labeled graphs

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GDM 2011


Graph similarity skyline 1 2
Graph Similarity Skyline (1/2)

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

GDM 2011


Graph similarity skyline 2 2
Graph Similarity Skyline (2/2)

  • q: a query, D = {g1, …, gn} a GDB

    • For i = 1 ton, 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) < Distk(g’, q).

GDM 2011


Illustrative example 1 2
Illustrative Example (1/2)

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Fig. 6– Graph databaseD and graph queryq

Fig. 6 – Graph databaseD and graph queryq

Tab. 3 – Information about |Mcs(gi, q)|

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GDM 2011


Illustrative example 2 2
Illustrative Example (2/2)

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

g1

g5

g1

Tab. 4 – Distance Measures

GSS(D, q) = {g1, g4, g5, g7}

GDM 2011


Refinement of graph similarity skyline 1 3
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

GDM 2011


Refinement of graph similarity skyline 2 3
Refinement of Graph Similarity Skyline (2/3)

  • Diversity of a subsetS of size kis:

    : diversity in the ith dimension of the subsetS

    s. t.:

GDM 2011


Refinement of graph similarity skyline 3 3
Refinement of Graph Similarity Skyline (3/3)

  • RefinementAlgorithm:

    • For j = 1 to , enumerate , with

    • For i = 1 to d, rank-order all Sj in decreasingwayaccording to theirdiversity

      Let be the rank of Sj w. r. t. the ith dimension:

      • : the best diversity value

      • : the worstdiversity value

  • EvaluateSj by:

  • Extract :

GDM 2011


Illustrative exa mple

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Illustrative Example

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  • Return the 2 best graphs:

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Fig. 8 – The skyline GSS

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GDM 2011


Summary and outlook
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

GDM 2011


Thank you

Thankyou

Questions ?