1. On Efficient Graph Substructure Selection
School of Computer Science and Engineering
Xiang Zhao† H. Shang§ W. Zhang † X. Lin† W. Xiao ‡
† The University of New South Wales, Australia
§ The University of Tokyo, Japan
‡ National University of Defense Technology, China

2. Outline Motivation Preliminaries Proposed Method Experiments
Conclusion 2

3. Single-labeled Graphs
Motivation
Single-labeled Graphs
Multi-attributed Graphs 3

4. Preliminaries
A data graph is a multi-attributed graph r = (Vr, Er, lr), where Vr is the vertex set, Er ⊆ Vr×Vr is the edge set, and lr is an attribute function.
v ∈ Vr has an attribute set Av, and each attribute a ∈ Av is assigned a value lr(a).
Note v may not have a particular attribute a, in which case lr(a) = nil. 4

5. Example 5

6. Preliminaries (Cont.)
A query graph is denoted by s = (Vs, Es, φs), where Vs is the vertex set, Es ⊆ Vs×Vs is the edge set, and φs is an attribute selection function.
Es enforces the connection constraints
φs exerts the attribute constraints.
v ∈ Vs has an attribute set Av such that a ∈ Av is assigned a selection condition φs(a). 6

7. Example 7

8. Preliminaries (Cont.)
Given u ∈ Vr, v ∈ Vs, u satisfies v on attribute a, provided
lr(a) = φs(a), if φs(a) defines an equality condition; or
lr(a) ∈ φs(a), if φs(a) defines a range condition; or
lr(a) ⊆ φs(a), if φs (a) defines a set containment condition; or
arbitrary value in the domain, if φs (a) is a wildcard.
u matches v, if u’s attribute values satisfies v’s corresponding condition conjunctively, for all v’s attribute constraints. 8

9. Problem Statement
Given a data graph r and a query graph s, a substructure mapping is an injection f : Vs → Vr such that
∀v ∈ Vs, f(v) ∈ Vr, f(v) matches v; and
∀(u, v) ∈ Es, (f(u), f(v)) ∈ Er.
Given a set of data graph as database R and a query graph s, the problem of substructure selection finds all substructure mappings from the query graph s to each data graph r in R. 9

10. Example 10

11. Basic Algorithmic Framework
Backtrack algorithm following DFS fashion
Order the query vertices into a sequence
Try to match query vertices iteratively
Test attribute and connection constraints
Success, match the next vertex in succession
Failure, backtrack and map the previous query vertex to another data vertex
Terminates when all full mappings are found 11

12. Example 12

13. Advanced Techniques Performance issue: Proposed optimization:
Not leverage any index
Naïve vertex-a-time extension
Proposed optimization:
Two-tier index and construction algorithm
Cost-based query processing algorithm 13

14. Substructure Selection Index
Template graphs
SS-index
Mappings 14

15. Primitive Operations Index retrieval Graph scan Attribute validation
Connection validation
Mapping extension
Mapping join
Given a substructure selection problem, finding the query execution plan with minimum cost is NP-hard. 15

16. Practical Plan Generation
Guidelines:
A proper execution order reduces the overall processing cost
Joining overlapped mapping reduces intermediate results
Eager constraints validation reduces intermediate results
Early validation of selective constraints reduces intermediate results
A cost-based heuristic algorithm
that iteratively selects the template graph
with minimum cost till all vertices are covered 16

17. Experimental Results on AIDS
Tree patterns provides the best balance between index size and runtime performance
As expected, SS-index builds index faster than GADDI, but slower than QuickSI 17

18. Experimental Results (cont.)
SS-search effectively reduces the query response time when query becomes larger, due to the effect of SS-index 18

19. Experimental Results (cont.)
SS-search performs the best when
Varying the percentage of vertices with attribute constraints
Varying the average range of selection conditions 19

20. Experimental Results (cont.)
SS-search scales well with large database, better than alternatives
SS-index can be as scalable as QuickSI if use only a portion of the database for mining frequent substructures 20

21. Conclusion
A new type of query substructure selection that handles general structure search on multi-attributed graphs
A novel structure SS-index to speed up the online computation via judiciously materializing partial embeddings
SS-search algorithm that dynamically composes effective query execution plan for efficient mapping discovery 21

22. Thank you!
Questions? 22

23. Related Work
Many indices for subgraph search, e.g., gIndex[SIGMOD 2004], FG-index[SIGMOD 2007], CT-index[ICDE 2011], CP-index[CIKM 2011]
Containment search or all-matching computation
Single-labeled graphs
DELTA[CIKM 2012] is the only work that deals with multi-labeled
Transform into a multi-dimensional indexing problem
Fixed attribute set 23