University of Illinois at Urbana-Champaign
Peixiang Zhao Jeffrey Xu Yu Philip S. Yu
CS@UIUC SEEM@CUHK IBM T. J. Watson Research Center
September 12 th , 2007
VLDB’07 Vienna, Austria

• Graph Containment Query
• Algorithmic Framework
• Indexability of frequent
Tree s
• Discriminative graph feature selection: Δ
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•
Graph is a mathematical construct and a general data structure representing relations among entities
• The emergence and the dominance of graphs asks for effective graph data management and mining tools so that users can organize, access, and analyze graph data efficiently
• Structural Pattern Mining:
Given a graph database, what are the potentially interesting structural patterns and how can we find them?
• Graph Indexing and Search:
How can we index graphs and perform searching, either exactly or approximately, in large graph databases?
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• Graph Containment Query
• Given a graph database G = { g
1
, g
2
, …, g
N
} and a query graph q , find the set  i
 g ,g i i

G }
• NP, since subgraph-isomorphism checking is NP-Complete
• Infeasible to check subgraph isomorphism sequentially for every g i in
G , especially challenging when graphs in G are large, or G is large and diverse
• Graph indexing!
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• Index construction generates the index feature set F from the graph database G . For each feature f , sup ( f ) is maintained
• Query processing is performed in a filtering-verification fashion:
• The filtering phase uses indexing features contained in q to compute the candidate answer set
Every graph in C q contains all q 's indexed features. Therefore, the query answer set, sup ( q ), is a subset of C q
• The verification phase checks subgraph isomorphism for every graph in
C q
. False positives are pruned and the true answer set sup ( q ) is returned
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• The cost of processing a graph containment query q upon G , denoted C , can be modeled as below
• C f
: the filtering cost
• C v
: the verification cost (NP-Complete)
• Analysis
1.
The key issue to improve query performance is to minimize |C q
|
2.
The indexing feature set F is quite relevant to C f and |C q
|
3.
Index construction performance: the feature selection cost C fs to construct F from among G
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•
Path -based Indexing approach
• All existing paths up to a certain length lp are enumerated as indexing features
– Index can be constructed efficiently
– Index size is quite large when lp is not small
– Limited pruning power, mainly because the structural information exhibited in graphs is lost when breaking graphs into paths
•
GraphGrep ( PODS’02 )
• Graph -based Indexing approach
• Subgraphs of G with different characteristics are selected as indexing features
– A costly index construction process
– Compact index structure
– Great pruning power, since structural information of graph is well-preserved
• gIndex ( SIGMOD’04 , PODS’05 ), C-Tree ( ICDE’06 ), GString ( ICDE’07 ), GDIndex
( ICDE’07 ), FG-Index ( SIGMOD’07 )
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Tree -
• Tree : Better indexability in comparison with path and graph
– The majority of frequent graph-features of
G are usually tree-features indeed
– Frequent tree-features and graph-features share similar distributions and frequent tree-features have similar pruning power like graph-features
– tree mining can be done much more efficiently than graph mining on G
• Δ : On-demand select a small number of discriminative graph-features without conducting costly graph mining beforehand
• Orders of magnitude smaller in index size, but performs much better than existing approaches in indexing construction and query processing
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• Frequent features (paths, trees, graphs) expose intrinsic characteristics of a graph database, G . They are representatives to discriminate between different groups of graphs in a graph database
• Which one should we index? Path, Tree or Graph?
1. The frequent feature set size: | F |
2. The feature selection cost: C fs
3. the candidate answer set size: |C q
|
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• Evidences:
• Among all frequent graph-features of G , a majority of them are trees indeed
– All subtrees of a frequent graph are frequent
– There is little chance that subtrees of frequent graph g coincide with those of frequent graph g ’ , due to the structural diversity and label variety
• Frequent paths share a very small portion, because a path-feature has a simple linear structure, which has little variety in structural complexity
• In terms of feature distributions, tree-features and graph-features share a very similar distribution w.r.t.
feature size
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The Real Dataset
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The Synthetic Dataset
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fs
• Given a graph database,
G , and a minimum support threshold,
σ , to discover the frequent feature set F ( F
P
/ F
T
/ F
G
) from G
Path Tree Graph
Isomorphism
Sub-Isomorphism
O( n )
O( n + m )
O( n ) P or NPC ( ?
)
O( m 3/2 n /log m ) NP-Complete
• Tree
• A good compromise between
– the more expressive, but computationally harder general graph
– the faster but less expressive path
• Specialization of general graph avoiding undesirable theoretical properties and algorithmic complexity incurred by graph
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q
|
• We define the pruning power power( f ) of a frequent feature f as
• The pruning power of a frequent feature set
S = { f
1
, f
2
, …, f n
}
•
Theorem 1 : Given a frequent graph-feature g , and let its frequent subtree set be T ( g ) = { t
1
, t
2
, …, t n
}. Then, power( g ) ≥ power( T ( g ))
• Theorem 2 : Given a frequent tree-feature t , and let its frequent sub-path set be P ( t ) = { p
1
, p
2
, …, p m
}. Then, power( t ) ≥ power( P ( t ))
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• The pruning power of all frequent subtree features,
T ( g ), of a frequent graph-feature g can be similar to the pruning power of g
• There is a big gap between the pruning power of a graphfeature g and that of all its frequent sub-path features, P ( g )
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The Real Dataset
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• It is feasible and effective to select F
T
, instead of F
G
, as indexing features for the graph containment query problem
• The frequent tree-feature set, F
T
, dominates F
G
• Discovering frequent tree-features from G can be done much more efficiently than mining frequent general graph-features
• F
T can contribute similar pruning power like that provided by F
G
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• Consider a query graph q which contains a subgraph g
• If power( T ( g )) ≈ power( g ), there is no need to index the graph-feature g , because its subtrees jointly have the similar pruning power
• if power( g ) >> power( T ( g )), it will be necessary to select g as an index feature because g is more discriminative than T ( g ), in terms of pruning
• Discriminative graph-features ( w.r.t.
its subtree-features, controlled by
ε
0
) are selected from queries on-demand, without mining the whole set of frequent graph-features from G beforehand
• Discriminative graph-features are used as additional indexing features, denoted Δ , which can also be reused further to answer subsequent queries
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• Given two graphs 
, where

• If the gap between power( g’ ) and power( g ) is large enough, g’ will be reclaimed from G ;
• Otherwise , g is discriminative enough for pruning purpose, and there is no need to reclaim g’ in the presence of g
• Approximate the discriminative computation between g’ and g , in the presence of our knowledge on frequent tree-features discovered
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• The occurrence probability of g in the graph database, G
• the conditional occurrence probability of g’
, w.r.t. g, models the probability to select g’ from G in the presence of g
• The upper and lower bound of
Pr ( g’
| g )
• The conditional occurrence probability of
Pr ( g’|g
), is solely upper-bounded by T ( g’
)
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• The Real Dataset
• The AIDS antiviral screen dataset from Developmental Theroapeutics
Program in NCI/NIH
• 42390 compounds retrieved from DTP's Drug Information System
• 63 kinds of atoms in this dataset, most of which are C, H, O, S, etc.
• Three kinds of bonds are popular in these compounds: single-bond, double-bond and aromatic-bond
• On average, compounds in the dataset has 43 vertices and 45 edges.
• The graph of maximum size has 221 vertices and 234 edges
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• The real dataset: false positive ratio (| Cq |/|sup( q )|) w.r.t.
the database size (= 1,000; 2,000; 4,000; 8,000; 10,000)
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• Generated by a widely-used graph generator, which is controlled by the following parameters :
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• Graph indexing plays a critical role in graph containment query processing on large graph databases
• Path-based and graph-based indexing approaches suffer from overly large index size, substantial index construction overhead and expensive query processing cost
• (Tree+Δ) is an effective and efficient graph indexing feature to answer graph containment queries
• (Tree+Δ) holds a compact index structure, achieves good performance in index construction and most importantly, provides satisfactory query performance for answering graph containment queries over large graph databases
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University of Illinois at Urbana-Champaign
VLDB’07 Vienna, Austria