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University of Illinois at Urbana-Champaign
Graph Indexing: Tree + Δ ≥ Graph
Peixiang Zhao
CS@UIUC
Jeffrey Xu Yu
SEEM@CUHK
Philip S. Yu
IBM T. J. Watson Research Center
September 12th, 2007
VLDB’07 Vienna, Austria
Synopsis
• Introduction
• Graph Containment Query
• Algorithmic Framework
• Related Work
• Tree + Δ
• Indexability of frequent Trees
• Discriminative graph feature selection: Δ
• Experimental Study
• Conclusion
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Introduction
• 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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Introduction
• Graph Containment Query
• Given a graph database G = {g1, g2, …, gN} and a query graph q, find
the set sup( q )  { gi | q  gi ,gi  G }
• NP, since subgraph-isomorphism checking is NP-Complete
• Infeasible to check subgraph isomorphism sequentially for every gi in
G, especially challenging when graphs in G are large, or G is large and
diverse
• Graph indexing!
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Graph Indexing: Algorithmic Framework
• 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 Cq contains all q's indexed features. Therefore,
the query answer set, sup(q), is a subset of Cq
• The verification phase checks subgraph isomorphism for every graph in
Cq. False positives are pruned and the true answer set sup(q) is returned
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Query Cost Model
•
The cost of processing a graph containment query q upon G,
denoted C, can be modeled as below
•
•
Cf : the filtering cost
•
Cv : the verification cost (NP-Complete)
Analysis
1.
The key issue to improve query performance is to minimize |Cq|
2.
The indexing feature set F is quite relevant to Cf and |Cq|
3.
Index construction performance: the feature selection cost Cfs to
construct F from among G
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Related Work
• 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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An alternative approach: (Tree + Δ)
• Tree-based Graph Indexing
• 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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Indexability of Path, Tree and Graph
•
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: Cfs
3. the candidate answer set size: |Cq|
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The Frequent Feature Set Size: | F |
• 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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Experiments on Two Datasets w.r.t. | F |
The Real Dataset
The Synthetic Dataset
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The feature selection cost: Cfs
• Given a graph database, G, and a minimum support threshold,
σ, to discover the frequent feature set F (FP / FT / FG ) from G
Path
Tree
Graph
Isomorphism
O(n)
O(n)
P or NPC (?)
Sub-Isomorphism
O(n + m)
O(m3/2n/logm)
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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The Candidate Answer Set Size: |Cq|
• We define the pruning power power(f) of a frequent feature f
as
• The pruning power of a frequent feature set S = {f1, f2 , …, fn}
• Theorem 1: Given a frequent graph-feature g, and let its frequent subtree set be T (g) = {t1, t2 , …, tn}. Then, power(g) ≥ power(T (g))
• Theorem 2: Given a frequent tree-feature t, and let its frequent sub-path
set be P (t) = {p1, p2 , …, pm}. Then, power(t) ≥ power(P (t))
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Pruning Power
• 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)
The Real Dataset
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Indexability of Path, Tree and Graph
• It is feasible and effective to select FT , instead of FG, as
indexing features for the graph containment query problem
• The frequent tree-feature set, FT , dominates FG
• Discovering frequent tree-features from G can be done much more
efficiently than mining frequent general graph-features
• FT can contribute similar pruning power like that provided by FG
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Discriminative Graph Features Δ
• 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 ε0) 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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Discriminative Graph Selection
• Given two graphs g, g’ q , where g g’
• 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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Discriminative Graph Selection
• 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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Experimental Studies
• 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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Experimental Studies
• The real dataset: index construction
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Experimental Studies
• 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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Experimental Studies
• The Synthetic Dataset
• Generated by a widely-used graph generator, which is
controlled by the following parameters:
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Experimental Studies
• The synthetic dataset: false positive ratio
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Conclusion
• 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
Thank you
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