Graph Substructure Search Xuemin Lin School of Computer Science and Engineering University of New South Wales Sydney, Australia

Graph Substructure Search

Xuemin LinSchool of Computer Science and Engineering

University of New South WalesSydney, Australia

Applications of Graphs• Chem-informatics Chemical Compounds (Small Size)

• Bio-informatics Protein Interaction Networks (Medium Size)

• Other Applications Social Networks (Large Size) … …

Fundamental Problems in Graph Database• Given a graph database D = {g1, ..., gn} of n data

graphs and a query graph q,

Substructure SearchRetrieve all data graphs which contain q.

Application : Chemical Compounds’ substructure Identification et al .

Supstructure Search Retrieve all data graphs which are contained by q.

Application : Molecule Function Prediction et al.

Substructure Search

Similarity Search?

Input Mistake

Exploration Queries

......

Outline

• Substructure Search (VLDB08)• Substructure Similarity Search

(SIGMOD10)• Superstructure Search (SSDBM10)• Superstructure Similarity Search

(ICDE2010)• Conclusions and Remarks

Substructure Search: gIndex (SIGMOD04) Index a set F of features from D.

, sup(f): set of graph ids in D contain f

Filtering:

Verification: verify each data graph in Cq.

sup( )q

f q f F

C f

Ff

gIndex (SIGMOD04)

q

g1 g2 g3

ID-List: {g1,g2}

Feature A:

PrunedPass (Feature A) Pass (Feature A)

False Positive Answer

Filtering:

Verification:

Tree+Delta (VLDB07)Objective: reduce the costs for building index

Observation: lower costs for generating tree-based features. most (95%) frequent subgraphs (in sparse graphs) are

trees.

Tree+Delta: • select frequent tree features, and then• add a small number of effective subgraphs.

FG Index (SIGMOD07)Objective: Index only query processing.

q is a frequent subgraph: if q is indexed, then return sup(q). if a supergraph q’ of q is indexed, no verification for sup

(q’).

q is not a frequent subgraph: # of verifications is bounded by

|| D

QuickSI (VLDB08): our workObjective: develop efficient verification algorithm speed up both verification and filtering.

QuickSI: An efficient verification algorithm.―Encode query graphs: terminate earlier.―Enforce connectivity―Three novel pruning techniques

Up to orders of magnitude speed up.

QuickSI

q

g

Depth-First Traversal

1

1

2

23 4

567

3 45

67

Forwarding

Backtracking

QuickSI

q

g

1

1

2

23

45 6

7

3 45

67


Forwarding

Backtracking

QuickSI

q

g

1

1

2

23 4

567

3 45

67


Forwarding

QuickSI

q

g

Access infrequent labels as early as possible


QuickSI

q

g

1

11 1

111 1

1

1

1



Synchronized Depth-First Traversal

QuickSI

q

g

1

1

2

12 2

222

Sparse Graph!2x5=10 possible matching pairs


Retain connectivity

QuickSI

q

g

1

1 2

1

2

2

Sparse Graph!ONLY 2 possible matching pairs


Retain connectivity


QuickSI

q

g

1

1

2

23 4

567

3 45

67


Effectively use degree information

Deg=3

Deg=2Stop here

2

2

Retain connectivity


QuickSI

q

g

1

1

2

23 4

567

3 45

67


Retain connectivity

Effectively use degree information

Deg=3

Deg=2Stop here

21

Deg=3Continue

Determine the access order for q.

Depth First Traversal

Experimental ResultsSettings

Notations Filtering Verification

GSI gIndex (SIGMOD ’04) QuickSI

SSI Swift Index (This Paper) QuickSI

FG FG Index (SIGMOD ’07)

AIDS Antiviral dataset, a popular benchmark, 43k chemical bonds, “C” “N” “O” are the most frequent labels.The data sets and query sets are same as in gIndex and FG Index

Experiments – Response Time

Construction Time

# of Features

Index Size

FG 167 1641 12.5M

GSI 146.6 3276 13M

SSI 26.6 462 5.5

Real dataset Large real dataset

Construction Time

# of Features

Index Size

FG 2188 7100 53.8M

GSI 306.2 4394 13M

SSI 170 922 11.8

Substructure Similarity Search – Grafil (SIGMOD07)

Maximum Common Subgraph MCS: Given g1 and g2, the common graph of g1 and g2 with the maximal number of edges, mcs(g1, g2).

Grafil: find all g in D s.t. (|q| - |mcs(q, g)|) ≤ σ

(|q| - |mcs(q, g)|): number of missing edges

Some variants...

Substructure Similarity SearchSubgraph Similarity

?

Connected Substructure Similarity Search: our work (SIGMOD10)

Maximum Connected Common Subgraph MCCS: Given g1 and g2, the connected common graph of g1 and g2 with the maximal number of edges, mccs(g1, g2).

dis (q, g) := |q| - |mccs(q, g)|

Goal: find all g in D s.t. dis(q,g) ≤σ

NP-Complete.

Filtering: triangular inequality?

dis(Q,D)+dis(D,F) ≥ dis(Q,F) dis(Q, D) ≥ dis(Q,F) – dis(D,F)

dis(Q,D)

dis(Q,F) dis(D,.F)

Query (Q) Feature(F) Data (D)

dis(Q,D)+dist(D,F) ≥ dist(Q,F) ?

1


Query (Q) Data (D)

1dis(Q,D)

Feature(F) Data (D)

1 2

dist(Q,D)+dist(D,F) ≥ dist(Q,F) ?

Similarity Search (triangular inequality)

2dist(F,D)


dis(Q,D)+dis(D,F) ≥ dis(Q,F) ---- HOLD!

2dis(Q,F)

Query (Q) Feature(F)

1 2 2

0 1 3

Query(Q) Feature(F) Data (D)

dist(Q,D)

dis(Q,D)+dis(D,F) ≥ dis(Q,F) X

Triangular inequality: not always hold

dist(D,F)dist(Q,F)

Connectivity Dominance

Connectivity Dominance:

The connectivity of mccs(g1, g2) dominates the connectivity of g2 if there is a subgraph isomorphic mapping F from mccs(g1, g2) to g2 such that if removing a set S of edges in mccs(g1, g2) causes mccs(g1, g2) disconnected, then removing F(S) always causes g2 disconnected.

Theorem. Given three graphs g1, g2, and g3, if the connectivity of mccs(g1, g2) dominates g2 or the connectivity of mccs(g2, g3) dominates g2, then dist(g1, g3) ≤ dist(g1, g2) + dist(g2, g3).

Remark: Linear Algorithm given embeeding.

dist(Q,F)+dist(F,D) ≥ dist(Q,D)

Validation Rule 1 – index onlydist(Q,F)+dist(F,D) ≤ => dist(Q,D) ≤(if mccs(Q, F) dominates F or mccs(F, D) dominates F)

dist(Q,D)+dist(D,F) ≥ dist(Q,F)

Pruning Rule 1:dist(Q,F)-dist(D,F)> => dist(Q,D)>(if mccs(D, F) dominates D)

dist(F,Q)+dist(Q,D) ≥ dist(F,D)

Pruning Rule 2:dist(F, D)-dist(F, Q)> => dist(Q,D)>(if mccs(F, Q) dominates Q)

Basic idea:

1. enumerate sub-spanning trees of query graph such that the # of missing edges ≤ ; try to terminate the algorithm as early as possible.2. sharing the enumeration costs by two ways: a. not enumerate every thing from scratch. b. once enumerated, keep enumerated spanning trees. – organized in a

binary tree to reduce storage space. 3. extend QucikSI [VLDB08].

Verification Algorithm

Experiments

Experiments

cIndex [VLDB’07]: super structure search• Filtering-Verification Framework Filter false results by a feature-based index:

exclusion based. Verify each candidate against the query graph.

DatabaseDatabase IndexIndex QueryQuery

FilteringFiltering(gc)

(gb)

(ga) fa

fb

fc

q

Filtered!Filtered!

Filtered!Filtered!

Candidate!Candidate!VerificationVerificationAnswer!Answer!

GPTree [EDBT’09]

• Enhanced Filtering-Verification Framework Share test cost in filtering and verification, respectively.

b a

c a a

b a

c a b

fa

(ga)

(gb)

b c

c a c(gc)

afb

c

b

a

a

a a b

(ga) (gb)

c

b

c

a

c

a

(gc)

sharingacross groups?

b

c

a c(fa) (fb)

Sharingbetween two phases?

sharingacross

suffixes?

PrefIndex [SSDBM10]

• Sharing-based Filtering-Verification Framework Share test cost in filtering and verification, respectively.

b a

c a a

b a

c a b

fa

(ga)

(gb)

b c

c a c(gc)

afb

c

b

a

a

a a b

(ga) (gb)

c

b

c

a

c

a

(gc)

sharingacross groups

b

c

a c(fa) (fb)

sharingbetween two phases

No sharingacross

suffixes

Computation Sharing Cost Model

• Cost Gain (Computation Sharing Benefits) Given k master groups of data graphs, assume that

1) all data graphs in each group Gi contain a master feature fi ( 1 ≤ i ≤ k );2) the subgraph isomorphism test from fi to a query graph q is costfi;

The total cost gain (computation sharing benefits) from each master group Gi can be represented as follows:

• Maximized Gain Cluster the database into a disjoint set of master groups such

that the total gain is maximized (NP-hard).

Efficiency Test• Database and Query Sets Database: AIDS10K; Query Sets: Q20, Q40, Q60 Q80, Q80+;

Superstructure Similarity Search: our work (ICDE10)

Given a q and a g, dis(q, g) = |g| − |mccs(q, g)|.

Superstructure Similarity Search: find all g from D such that dis(q, g) ≤ σ.

Note: dis(q, g) = |q| − |mccs(q, g)| in substructure similarity search.

Observations: filtering framework in SIGMOD10 is immediately applicable. techniques in SIGMOD10 may not be effective for a nearly “super-containment” relationship. Sharing is possible just like PrefIndex.

SG-Enum Index (ICDE2010)Key Ideas:1. For a g, enumerate all subgraphs with at most σ edges removed,

σ-missing subgraphs.2. dis(q, g) ≤σ if and only if q contains a σ-missing subgraph.

Key issues:

Automorphic subgraphs?

Prefix-sharing? in one g among different data graphs

Query processing?

SG-Enum Index (ICDE2010)Top-down Construction:1. Enumerate all σ-missing subgraphs. 2. Iteratively, choose an edge as follows:

a) Always select an edge contained by most σ-missing subgraphs.b) Split the group into 2: one contain the edge and another does not contain the

edge.

Bottom-up Construction:1. Generate a sequence for each σ-missing subgraph.2. Merge the prefixes by chance.

Bottom-up among data graphs.

Query algorithm: extends QuickSI.

Experiments – Query Response Time

Conclusion and RemarksSubstructure search and its similarity searchSuperstructure search and its similarity search.(VLDB08, ICDE10, SIGMOD10, SSDBM10)

Issues: Similarity measures? Large data graphs?

Documents

Graph Substructure Search Xuemin Lin School of Computer Science and Engineering University of New South Wales Sydney, Australia