Download Slides - FSU Computer Science

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Extensible Storage Engine wikipedia, lookup

Clusterpoint wikipedia, lookup

Commitment ordering wikipedia, lookup

Serializability wikipedia, lookup

Database model wikipedia, lookup

Transcript
University of Illinois at Urbana-Champaign
Graph Indexing: Tree + Δ ≥ Graph
Peixiang Zhao
[email protected]
Jeffrey Xu Yu
[email protected]
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
VLDB’07 Vienna, Austria
2 of 25
Sept. 12th, 2007
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?
VLDB’07 Vienna, Austria
3 of 25
Sept. 12th, 2007
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!
VLDB’07 Vienna, Austria
4 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
5 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
6 of 25
Sept. 12th, 2007
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)
VLDB’07 Vienna, Austria
7 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
8 of 25
Sept. 12th, 2007
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|
VLDB’07 Vienna, Austria
9 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
10 of 25
Sept. 12th, 2007
Experiments on Two Datasets w.r.t. | F |
The Real Dataset
The Synthetic Dataset
VLDB’07 Vienna, Austria
11 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
12 of 25
Sept. 12th, 2007
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))
VLDB’07 Vienna, Austria
13 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
14 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
15 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
16 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
17 of 25
Sept. 12th, 2007
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’)
VLDB’07 Vienna, Austria
18 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
19 of 25
Sept. 12th, 2007
Experimental Studies
• The real dataset: index construction
VLDB’07 Vienna, Austria
20 of 25
Sept. 12th, 2007
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)
VLDB’07 Vienna, Austria
21 of 25
Sept. 12th, 2007
Experimental Studies
• The Synthetic Dataset
• Generated by a widely-used graph generator, which is
controlled by the following parameters:
VLDB’07 Vienna, Austria
22 of 25
Sept. 12th, 2007
Experimental Studies
• The synthetic dataset: false positive ratio
VLDB’07 Vienna, Austria
23 of 25
Sept. 12th, 2007
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
VLDB’07 Vienna, Austria
24 of 25
Sept. 12th, 2007
University of Illinois at Urbana-Champaign
Thank you
VLDB’07 Vienna, Austria