Computer Science and Engineering

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• Efficient Subgraph Similarity All-Matching

• Computer Science and Engineering

• Gaoping Zhu, Ke Zhu, Wenjie Zhang, Xuemin Lin,
  Chuan Xiao
• The University of New South Wales

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Outline

• Introduction
• Preliminary
• Framework
• Algorithms
• Experiments
• Conclusions

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Introduction Graph Data

• Chem-informatics
• Chemical Compounds (Small Size)
• Bio-informatics
• Protein Interaction Networks (Medium Size)
• Internet
• World Wide Web (Large Size)

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Introduction Subgraph All-Matching

• Problem
• Subgraph exact all-matching enumerates all exact
  matches of a query graph q in a large data graph
  G.
• Subgraph similarity all-matching enumerates all
  similarity matches of a query graph q in a large
  data graph G.
• Motivations
• Noisy query graphs due to erroneous user input.
• Noisy data graphs due to imprecise collection.

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Preliminaries

• Edge Edit Distance
• The edge edit distance from a graph g1 to another
  graph g2 is the minimum number of edge insertions
  required to transform g1 to g2.
• GED (p1, q) 0, GED (p2, q) 1.

B
B
B
B
B
B
A
C
A
C
A
C
q
p1
p2
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Preliminaries

• Feasible Pattern
• Given a distance threshold d, p is called a
  feasible pattern of q if p is a connected
  subgraph of q with no missing vertex and GED (p,
  q) d.
• The feasible patterns of q are p1, p2, p3, p4
  for d 1.

B
B
B
B
B
B
B
B
B
B
A
C
A
C
A
C
A
C
A
C
q, d 1
p1
p2
p3
p4
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Preliminaries

• Similarity Matches
• A similarity match of q in G is a subgraph
  isomorphic mapping from any feasible pattern p to
  q.
• Must consider any feasible pattern!
• Exact matches of q in G are also similarity
  matches!

similarity match
similarity match
B
C
A
B
B
B
B
B
G
A
C
A
C
q
p1
p2
exact match
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SAPPER VLDB10
Enumerate Phase
Search Phase
Results
Mp1
d 1
Mp2
G


Mp5
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Motivation I Effective Search Order
A
1 match
1 match
1 match
1 match
4 matches
12 matches
27 matches
Search Order One v1, v2, v3, v4, v5, v6, v7,
v8 ? 47 intermediate matches
A
C
B
B
B
B
B
B
Search Order Two v4, v3, v5, v6, v2, v1, v7,
v8 ?350 intermediate matches
D
B
G
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Motivation II Sharing Computation
p
p'
Query Execution Plan One search p and p
separately ? Share no computation
Query Execution Plan Two search f1, f2 and f2
and then join ?Share the computation on f1
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Framework - DecQ

• Query Decomposition (Phase One)
• Decompose the query graph q into a set of
  selective edge-disjoint sub-queries Q f1, ,
  fn , called fragments.

q
Query Graph
Decompose
Fragments
f1
f2
f3
f4
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Framework - DecQ

• Local Matching (Phase Two)
• Enumerate all local (feasible) patterns f of
  each fragment f and apply depth-first search on
  each pattern f to obtain the local matches
  (exact matches of f in G).

f
Fragments
Enumerate
fa
f'b
f'c
f'd
Local Patterns
Depth-first Search
Mfa
Mfb
Mfc
Mfd
Local Matches
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Framework - DecQ

• Global Matching (Phase Three)
• Enumerate all global (feasible) patterns p and
  merge the local matches of decomposed local
  patterns of p to obtain the global matches (exact
  matches of p in G).

p
Mp
Global Matches
Merge
Mf1
Mf3
Mf4
Mf2
Local Matches
Retrieve
f'1
f'2
f'3
f'4
Local Patterns
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Algorithms

• Local matching
• Enumerate all local patterns f of each fragment
  f.
• Search all exact matches of each f by
  depth-first search fashion with effective search
  order.
• Effective Search Order
• It is NP-complete to find an search order with
  minimum number of intermediate matches produced
  in the depth-first search.

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Algorithms

• Estimating Exact Matches of a Graph
• Given a graph f, assume M(v) / M(e) contains all
  mappings in G of a vertex v / edge e in f.
• For each edge (u, v) in f, given any u in M(u)
  and v in M(v), the probability that there is an
  edge (u, v) in G is
• The estimated number of exact matches of f in G
  can be represented by

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Algorithms

• Approximating Optimal Search Order
• A search order grow a local pattern f vertex by
  vertex.
• Greedy heuristic select the vertex v such that
  the number of estimated exact matches of the
  current subgraph s of f is minimized.

s3
f
s1
s2
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Algorithms

• Global Matching
• A global pattern p can be either a minimal or a
  non-minimal pattern.
• A minimal pattern p does not have one subgraph
  p, which is also a global pattern with one
  missing edge in p.
• A non-minimal pattern p has at least one subgraph
  p, which is also a global pattern with one
  missing edge in p.

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Algorithms

• Processing Minimal Patterns
• For a minimal pattern p, we decompose p into a
  set of local patterns and merge the local matches
  to obtain global matches Mp.

p'
p
store the matches of (f3 ? f4)
reuse the matches of (f3 ? f4)
f'1
f'2
f'3
f'4
f1
f2
f'3
f'4
M1
M3
M4
M2
M1
M3
M4
M2
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Algorithms

• Processing Non-minimal Patterns
• For a non-minimal pattern p, we pick the child
  pattern p of p with the smallest Mp. We check
  if the missing edge exists in each exact match of
  p in G. If so, this match is validated as an
  exact match of p in G.

B
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B
B
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A
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A
C
p
p
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Algorithms

• Decomposition Query Execution Plan
• Each decomposition of a global pattern p
  corresponds to a query execution plan of p.
  (i.e., as in RDBMS)
• It is costly to generate a good query execution
  plan for each global pattern p of q .
• Recursive Bisection
• We use heuristic solution to recursively bisect q
  into a set Q of edge-disjoint fragments.
• Bisect a graph into two subgraphs such that their
  graph size are balanced.

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Experiments

• Real Data
• Data Graph HPRD (Human Protein Interaction
  Network, V(G) 9,460 vertices, E(G) 37,081
  with vertices labeled by GO Term)
• Query Graphs selected subgraphs from HPRD
  network with 1-3 inserted noisy edge.
• Synthetic Data
• Data Graphs obtained by synthetic graph
  generator
• Query graphs selected subgraphs from data
  graphs with 1-3 inserted noisy edge.

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Experiments

• Evaluated Algorithms
• SAPPER
• ROND (Random search Order No Decomposition)
• EOND (Effective search Order No Decomposition)
• DecQ (Effective Search Order and Decomposition)
• Default Settings
• E(q) 40, avg. deg(q) 4
• E(G) 5k, avg. deg(G) 12, SL 100
• d 2

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Experiments

• Varying Error Threshold

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Experiments

• Varying Query Settings

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Experiments

• Varying Data Graph Settings

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Experiments

• Comparing with SAPPER

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Conclusions

• A novel framework DecQ for subgraph similarity
  all-matching.
• Effective search order for local matching with
  depth-first search fashion.
• Effective query decomposition plan for global
  matching with computation sharing.

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• Thank You!
• Any Questions?