Computer Science and Engineering

TreeSpanEfficiently Computing Similarity All-Matching

Gaoping Zhu#, Xuemin Lin#, Ke Zhu#, Wenjie Zhang#, Jeffrey Xu Yu†

# The University of New South Wales† The Chinese University of Hong

Kong

Outline

• Introduction• State-of-the-Art• Our Approach• Experiments• Conclusions

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Introduction — Graph Data

Chem-informatics Chemical Compounds (small size)

Bio-informatics PPI Networks (medium size)

Internet World Wide Web (large size)

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Introduction — Exact All-Matching (I) Exact All-Matching Enumerate all exact (i.e. isomorphic) matches

of a query graph q in a data graph G.

Applications Query biological patterns in PPI networks. Detect suspicious bugs in software programs.

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Introduction — Exact All-Matching (II) Dilemma of Exact All-Matching If q is issued by user for exploratory purpose … If G is noisy due to imprecise data collection …

Potential Solutions Modify q/G and run exact all-matching again and

again. Ask system to return approximate results (i.e.,

similarity all-matching)

No exact matches can be found!

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SAPPER [VLDB’10 Zhang et al] (I)

Similarity All-Matching Given a query graph q, a data graph G and a similarity

threshold θ, enumerate all similarity matches of q in G (i.e., all connected subgraphs of G missing at most θ edges in q).

Framework Enumerate a set of seeds QSAPPER (i.e., all connected

subgraphs q’ of q missing θ edges in q). Exact all-matching on each seed q’ to obtain exact

matches. Induce similarity matches based on exact matches of seeds.

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SAPPER [VLDB’10 Zhang et al] (II)

Cost Model

|QSAPPER | = # of exact all-matching tests

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Our Approach — Overview (I)

Tree-based Spanning Search Paradigm — TSpan Enumerate a set of seeds QT (i.e., spanning trees of q

cover all connected subgraph q’ of q missing θ edges in q).

Primary Contribution Reduce # of exact all-matching tests (i.e., # of seeds). Reduce the complexity of exact all-matching test from

graph to graph to tree to graph.

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more SAPPER seeds

3 all-matching tests on connected subgraphs of q 1 all-matching tests on a spanning tree of q

Our Approach — Overview (II)

Generating Similarity Maximal Matches

Generating similarity maximal matches only can reduce # of exact all-matching tests.

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F2 = {u1→v2, u2→v1, u3→ v3, u4→v4}similarity maximal matches

Our Approach — Problem Statement Similarity Maximal All-Matching Given a query graph q, a data graph G and a

similarity threshold θ, enumerate all distinct similarity maximal matches of q in G conforming θ.

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Our Approach — Seeding (I)

PRIM Order on Spanning Trees Similar to the basic idea of minimum spanning tree. Given a total order on E(q), a spanning tree T =

{T[0], T[1], …, T[|V(q)|- 1]} of q conforms PRIM order (T[0] is head vertex), if and only if each spanning edge T[i] has the smallest order in E(q) – {T[1], ..., T[i − 1]} and connects {T[0], T[1], ..., T[i − 1]}.

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Our Approach — Seeding (II)

Avoid Duplicate Results Two spanning trees of q may induce duplicate similarity

maximal matches.

Associate an edge exclusion set T.R to each T in QT.

T.R is a set of edges in E(q) – E(T) enforced to be mismatched in the similarity maximal matches induced by T.

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Our Approach — Seeding (III)

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QT Enumeration Algorithm

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Our Approach — Seeding (IV)

QT Enumeration Algorithm

Correctness : Using QT to inducing similarity maximal matches neither generates duplicate results nor misses valid results.

Minimality of QT : Missing any spanning tree in QT does not guarantee the completeness of results based on edge exclusion semantics.

When |E(q)| = m, |V(q)| = n, (1)|QSAPPER| ≥ |QT|;

(2) |QT| = |QSAPPER| only when θ = 0 or m − n + 1.)|.|

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Our Approach — Searching (I)

Effectively Storing QT

Use DFS Traversal Tree to share computation cost.e1e2

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Our Approach — Searching (II)

Similarity Maximal All-Matching Algorithm Sketch Traverse the DFS Traversal Tree in a depth-first backtrack

search fashion. go-down : Beginning from the initial spanning tree,

recursively drill down to extend the current partial match to the next spanning edge T[i] in the current spanning tree T.

alternate : If T[i] can not be extended based on the current partial match and we can still afford to mismatch T[i] by conforming θ, alternate the algorithm from T to the alternative spanning tree T’ enumerated by replacing T[i] with T’[i] .

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Our Approach — Optimizations

Optimizations (I) EnumrateOnDemand Strategy Motivation : further reduce the number of seeds. Enumerate an alternative tree T’ based on the current

tree T only when it is feasible to extend the current partial similarity maximal match conforming θ (1) on the next spanning edge T[i] or (2) on the next spanning edge T[i]’.

Optimizations (II) Effective Search Order Motivation : terminate all-matching test as early as

possible. Decide the search order of spanning edges in T based on

the post-filtering candidate sets of each vertex in q.

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Our Approach — Filtering & Ordering (I)

Neighborhood Aggregate N(v, g) Given a set of labels ΣV = {L1, ..., Lm}, N(v, g) = (x1,

..., xm) where xi is the number of neighbors of v in g with label Li ∈ ΣV.

Neighborhood-based Filtering Compute the candidate set C(u) for each u in q.

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N(u, q) = {2, 1, 0, 2}

N(v, G) = {1, 2, 2, 0}

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Our Approach — Filtering & Ordering (II)

QI Search Ordering [VLDB’08 Shang et al.] Pick Head Vertex : The vertex u in q with minimum φ(u) (i.e., the

occurrence of vertices in G with l(u)). Pick Next Spanning Edge : The edge (u1, u2) with minimum φ(u1,

u2) (i.e., the occurrences of edges in G with (l(u1), l(u2))) where u1 is a vertex incident on previous picked spanning edges.

Filtering-based Search Ordering Pick Head Vertex : The vertex u in q with minimum number of

candidates (i.e., |C(u)|). Pick Next Spanning Edge : The edge (u1, u2) minimizing

|C(u2)|×φ (u1, u2)/φ(u2)

where u1 is vertex incident on previous picked spanning edges.

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Experiments — Experimental Settings

Data Graphs GH : HPRD network (|V(GH)| = 9,460, |E(GH)| = 37,081).

GS : default synthetic data graph. Other synthetic data graphs generated by varying data graph settings.

Query Graphs Random selected subgraphs of the corresponding data graphs.

Parameter Settings (default settings in bold)

|V(G)| 5k, 10k, 20k, 40k, 80k

avg. deg(G)

4, 8, 12, 16, 20

|ΣV | 20, 50, 100, 200

|V(q)| 20, 40, 60, 80, 100

avg. deg(q) 3, 4, 5, 6

θ 1, 2, 3, 4

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|QSAPPER| : # of exact all-matching tests by SAPPER [VLDB’10].

|QT| : # of exact all-matching tests by EnumerateAll paradigm. TSpan : # of exact all-matching tests by EnumerateOnDemand

paradigm.

Experiments — # of exact all-matching tests

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Similarity All-Matching SAPPER : Generate all similarity matches. TSpan+ : Run TSpan first and then generate all similarity

matches based on similarity maximal matches.

Similarity Maximal All-Matching NaïveTSpan : Similarity maximal all-matching with no

computation sharing. TSpan : Similarity maximal all-matching with computation

sharing.

Experiments — Total Processing Time

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Enumeration Paradigms PrecTSpan : Similarity maximal

all-matching by EnumerateAll. TSpan : Similarity maximal all-

matching by EnumerateOnDemand.

Filtering & Ordering TSpanQI : TSpan algorithm with

QI searching ordering. TSpanNF : TSpan algorithm

with no filtering technique.

Experiments — Total Processing Time

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TSpan on Large-scale Datasets

Experiments — Large-scale Data Graphs

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Conclusions

Tree-based Spanning Search Paradigm

EnumerateOnDemand Strategy Filtering-based Search Ordering

SAPPER TSpan

# of all-matching tests

significantly less

each all-matching test

graph to graph tree to graph

computation-sharing no yes

similarity results non-maximal maximal

mC

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Thank You!Any Questions?