Graph Matching
Simulation based approach
Shang Zechao
1010161920
Introduction
• What is graph matching?
• When the one graph matches with
another?
Introduction (cont.)
• Graph: G=(V, E). GQ = (VQ, EQ)
• Can be easily extended with labels.
• Exact matching: isomorphism
– Find a bijection function f between V and VQ
– (u, v) in E iff (f(u), f(v)) in EQ
Introduction (cont.)
• Graph isomorphism
– GI class
• Sub-graph isomorphism
– NP-Complete
• Too hard!
Simulation based approach
[Henzinger95]
• Find a relation S: V x VQ
• (u, u’) in S if
– u and u’ has same labels
– for all children v’ of u’, there exists v
•
•
V is child of u
(v, v’) in S
Simulation based approach
• The major difference between graph
simulation and graph isomorphism
– Isomorphism requires an bijection (one to
one) function
– Graph simulation based on relation (many to
many)
• Simulation is in polynomial time
An Example [Fan10]
• Drug dealer network
– B: Boss
– S: Secretary
– AM: Assistant manager
– FW: Field worker
An Example (cont.)
• In real world
– S and AM is same
– AM maps to multiple worker
Bounded Simulation [Fan10]
• Each edge in pattern graph has label
– Either a positive integer K
– Or * (infinite)
• The length of path connects these two
nodes
The Example (cont.)
• AM should be able to reach FW within 3
hops.
Matching Algorithm
• Similar with the EffcientSimilarity algorithm
in [Henzinger95].
– Pre-compute the distance matrix between all
pairs of node in G.
• Complexity O(|V||E| + |Ep||V|2 + |Vp||V|)
Strong Simulation [Ma12]
• Recall the condition that two nodes match:
– Have same label
– Children could be matched by simulation
• Two issues
– Parent information is not captured
– Matching size is not limited
An Example [Ma12]
• Bio can match to Bio1, Bio2, Bio3, Bio4
– Actually only Bio4 makes sense
Strong Simulation
• two nodes match if:
– Have same label
– Children could be matched by simulation
– Parent could be matched by simulation
• The matched sub-graph should have
same diameter as pattern graph
An Example (cont.)
• Bio only matches to Bio4 in strong
simulation
Comparison of different
approaches
simulation
children parents connect cycle
topology topology ivity
info
Y
N
N
N
with parent
topology
Y
Y
Y
Y
with diameter
constrain
Y
Y
Y
Y
isomorphism
Y
Y
Y
Y
Comparison of different
approaches
simulation
locality bounded bisimula bounded
matches tion
cycle
N
Y
N
N
with parent
topology
N
N
N
N
with diameter
constrain
Y
Y
N
N
isomorphism
Y
N
Y
Y
But
• Bounded cycle problem is intractable
– NP-hard
• Bisimilar problem is intractable
– coNP-hard
References
• [Henzinger95] M. R. Henzinger, T. A. Henzinger, and P.
W. Kopke. 1995. Computing simulations on finite and
infinite graphs. In Proceedings of the 36th Annual
Symposium on Foundations of Computer
Science (FOCS '95). IEEE Computer Society,
Washington, DC, USA, 453-.
• [Fan10] Wenfei Fan, Jianzhong Li, Shuai Ma, Nan Tang,
Yinghui Wu, and Yunpeng Wu. 2010. Graph pattern
matching: from intractable to polynomial time. Proc.
VLDB Endow. 3, 1-2 (September 2010), 264-275.
• [Ma12] Shuai Ma, Yang Cao, Wenfei Fan, Jinpeng Huai ,
Tianyu Wo. 2012. Capturing Topology in Graph Pattern
Matching. PVLDB. To appear.