1. iGraph: A Framework for Comparisons of Disk-Based Graph Indexing Techniques Jeffrey Xu Yu et. al. VLDB ‘10 Presented by Tao Yu

2. Why I choose this paper Disk-based Implementation technique Graph database Application Dataset iGraph: A Framework for Comparisons of Disk-Based Graph Indexing Techniques Jeffrey Xu Yu et. al. VLDB ‘10 Presented by Tao Yu

3. Why I choose this paper Disk-based Implementation technique Graph database Application Dataset

4. Why they write this paper Provide a uniform test framework. Binary executable wall clock time comparison is not fair. Some algorithms are in-memory implemented while others are on-disk implemented. Obtain real disk I/Os by bypassing OS disk cache. Perform a large number of tests. Why I choose this paper Disk-based Implementation technique Graph database Application Dataset

5. Why they write this paper Provide a uniform test framework. Binary executable wall clock time comparison is not fair. Some algorithms are in-memory implemented while others are on-disk implemented. Obtain real disk I/Os by bypassing OS disk cache. Perform a large number of tests.

6. Background Application Graph isomorphism Stream A large number of small graphs Undirected labeled graph G1 = (V;E;Lv;L1e) Why they write this paper Provide a uniform test framework. Binary executable wall clock time comparison is not fair. Some algorithms are in-memory implemented while others are on-disk implemented. Obtain real disk I/Os by bypassing OS disk cache. Perform a large number of tests.

7. Background Application Graph isomorphism Stream A large number of small graphs Undirected labeled graph G1 = (V;E;Lv;L1e)

8. Related work Mining based approaches Non-mining based approaches Size : #Edges Background Application Graph isomorphism Stream A large number of small graphs Undirected labeled graph G1 = (V;E;Lv;L1e)

9. Related work Mining based approaches Non-mining based approaches Size : #Edges

10. FG-Index Indexing All frequent subgraphs All infrequent edges Query Enumerate a subset of subgraphs Verification-free strategy gIndex Indexing All frequent subgraphs (maxL) A subset of infrequent subgraphs (maxL) Discrimitive features Query Enumerate all subgraphs (maxL) Related work Mining based approaches Non-mining based approaches Size : #Edges

11. FG-Index Indexing All frequent subgraphs All infrequent edges Query Enumerate a subset of subgraphs Verification-free strategy gIndex Indexing All frequent subgraphs (maxL) A subset of infrequent subgraphs (maxL) Discrimitive features Query Enumerate all subgraphs (maxL)

12. SwiftIndex Indexing All frequent trees size up to maxL All discriminative trees size up to maxL All infrequent edges Query PrefixQuickSI SwiftIndex Indexing All frequent trees size up to maxL All discriminative trees size up to maxL All infrequent edges Query PrefixQuickSI Tree+Δ Indexing All frequent trees size up to maxL – 1 All infrequent edges Generates graph features on the fly Query Enumerate all subtrees (maxL) FG-Index Indexing All frequent subgraphs All infrequent edges Query Enumerate a subset of subgraphs Verification-free strategy gIndex Indexing All frequent subgraphs (maxL) A subset of infrequent subgraphs (maxL) Discrimitive features Query Enumerate all subgraphs (maxL) Tree+Δ Indexing All frequent trees size up to maxL – 1 All infrequent edges Generates graph features on the fly Query Enumerate all subtrees (maxL)

13. SwiftIndex Indexing All frequent trees size up to maxL All discriminative trees size up to maxL All infrequent edges Query PrefixQuickSI Tree+Δ Indexing All frequent trees size up to maxL – 1 All infrequent edges Generates graph features on the fly Query Enumerate all subtrees (maxL)

14. C-Tree Indexing A hierarchical tree of graph closure Query Pseudo subgraph isomorphism test SwiftIndex Indexing All frequent trees size up to maxL All discriminative trees size up to maxL All infrequent edges Query PrefixQuickSI Tree+Δ Indexing All frequent trees size up to maxL – 1 All infrequent edges Generates graph features on the fly Query Enumerate all subtrees (maxL) GraphGrep Indexing All paths (maxL) Query Enumerate all paths (maxL)

15. C-Tree Indexing A hierarchical tree of graph closure Query Pseudo subgraph isomorphism test GraphGrep Indexing All paths (maxL) Query Enumerate all paths (maxL)

16. Isomorphism Algorithms VF2 QuickSI C-Tree Indexing A hierarchical tree of graph closure Query Pseudo subgraph isomorphism test GraphGrep Indexing All paths (maxL) Query Enumerate all paths (maxL) gCode Indexing Vertex signature from neighbors Graph signature from vertex GCode-Tree Query Index level (graph signature) Object level (vertex signature)

17. Isomorphism Algorithms gCode Indexing Vertex signature from neighbors Graph signature from vertex GCode-Tree Query Index level (graph signature) Object level (vertex signature) VF2 QuickSI

18. Implementation Graph A list of vertices and a list of edges If a graph is less than the page size Store it as a tuple in a heap page Else Store it as a BLOB B+-tree for all graphs by graph ID Other techniques CAM code to encode feature Djb2 hash function Mini-page Isomorphism Algorithms gCode Indexing Vertex signature from neighbors Graph signature from vertex GCode-Tree Query Index level (graph signature) Object level (vertex signature) VF2 QuickSI

19. Implementation Graph A list of vertices and a list of edges If a graph is less than the page size Store it as a tuple in a heap page Else Store it as a BLOB B+-tree for all graphs by graph ID Other techniques CAM code to encode feature Djb2 hash function Mini-page

20. Dataset Small sparse AIDS: 10000 graphs 25.42 vertices and 27.40 edges 51 vertex lables and 4 edge labels Small dense GraphGen: 10000 graphs 7 vertices and 30 edges 20 vertex lables and 20 edge labels Large PubChem: 1000000 graphs 23.98 vertices and 25.76 edges 81 vertex lables and 3 edge labels Implementation Graph A list of vertices and a list of edges If a graph is less than the page size Store it as a tuple in a heap page Else Store it as a BLOB B+-tree for all graphs by graph ID Other techniques CAM code to encode feature Djb2 hash function Mini-page

21. Dataset Small sparse AIDS: 10000 graphs 25.42 vertices and 27.40 edges 51 vertex lables and 4 edge labels Small dense GraphGen: 10000 graphs 7 vertices and 30 edges 20 vertex lables and 20 edge labels Large PubChem: 1000000 graphs 23.98 vertices and 25.76 edges 81 vertex lables and 3 edge labels

22. Query sets For AIDS: the existing query sets Q4, Q8, · · ·, Q24 can be downloaded from [3]. Each query set Qn contains 1000 graphs where each graph size is n. For the other datasets: First, randomly select 1000 graphs from each dataset whose size is larger than or equal to 24. Then, for each graph g, we remove edges until g is still connected and contains 24 edges. This query set is called Q24. In order to generate Q20, we remove edges from each graph in Q24 until the remaining graph contains 20 edges. We repeat this process to generate the remaining query sets. Dataset Small sparse AIDS: 10000 graphs 25.42 vertices and 27.40 edges 51 vertex lables and 4 edge labels Small dense GraphGen: 10000 graphs 7 vertices and 30 edges 20 vertex lables and 20 edge labels Large PubChem: 1000000 graphs 23.98 vertices and 25.76 edges 81 vertex lables and 3 edge labels

23. Query sets For AIDS: the existing query sets Q4, Q8, · · ·, Q24 can be downloaded from [3]. Each query set Qn contains 1000 graphs where each graph size is n. For the other datasets: First, randomly select 1000 graphs from each dataset whose size is larger than or equal to 24. Then, for each graph g, we remove edges until g is still connected and contains 24 edges. This query set is called Q24. In order to generate Q20, we remove edges from each graph in Q24 until the remaining graph contains 20 edges. We repeat this process to generate the remaining query sets.

24. Disk schedule LRU as buffer replacement algorithm Page size: 8 K FILE_FLAG_NO_BUFFERING Query sets For AIDS: the existing query sets Q4, Q8, · · ·, Q24 can be downloaded from [3]. Each query set Qn contains 1000 graphs where each graph size is n. For the other datasets: First, randomly select 1000 graphs from each dataset whose size is larger than or equal to 24. Then, for each graph g, we remove edges until g is still connected and contains 24 edges. This query set is called Q24. In order to generate Q20, we remove edges from each graph in Q24 until the remaining graph contains 20 edges. We repeat this process to generate the remaining query sets.

25. Disk schedule LRU as buffer replacement algorithm Page size: 8 K FILE_FLAG_NO_BUFFERING

26. Experiment The database construction cost of gIndex is comparable to all feature selectionmethods such as Tree+∆, FG-Index, and SwiftIndex. “We have communicated with Xifeng Yan who first ignored the edge label. He did that simply in order to “make the problem more difficult.” Subsequent work imitated his setting without clear reason.”

27. Experiment More features, less candidates. The gIndex performs the best.

28. Experiment For Q4, FG-Index performs the best since it exploits the verification-free strategy. gCode performs the worst: 1) more candidates 2) lookups over the vertex signature dictionary need more buffering. gCode Indexing Vertex signature from neighbors Graph signature from vertex GCode-Tree Query Index level (graph signature) Object level (vertex signature)

29. Experiment As for C-Tree, the number of disk I/Os is slightly reduced compared with a small buffer size, since the database size of C-Tree is still larger than the buffer size, and tree traversal incurs the sequential flooding effect C-Tree Indexing A hierarchical tree of graph closure Query Pseudo subgraph isomorphism test

30. Experiment gIndex is slightly slower than FG-Index and SwiftIndex due to slow subgraph enumeration from a query. This fact indicates that the I/O cost must be carefully optimized to obtain good performance. gIndex Indexing All frequent subgraphs (maxL) A subset of infrequent subgraphs (maxL) Discrimitive features Query Enumerate all subgraphs (maxL)

31. Experiment Only 37 frequent features. Almost all features in FG- Index, Tree+∆, and SwiftIndex are infrequent features. gCode use signatures. gIndex mines all infrequent and discriminative features of size up to 3. gIndex Indexing All frequent subgraphs (maxL) A subset of infrequent subgraphs (maxL) Discrimitive features Query Enumerate all subgraphs (maxL) gCode Indexing Vertex signature from neighbors Graph signature from vertex GCode-Tree Query Index level (graph signature) Object level (vertex signature)

32. Experiment Drastic changes to gCode (I), C-Tree (I), and Tree+∆. Frequent feature space is small. Graph features reclaimed at small sizes are used for larger query sizes. Tree+Δ Indexing All frequent trees size up to maxL – 1 All infrequent edges Generates graph features on the fly Query Enumerate all subtrees (maxL)

33. Experiment FG-Index does not outperform gIndex even for Q4 since there exist no frequent features of size 4. Queries in this dense synthetic dataset contain many cycles, and thus, the cost of mining graph features on the fly is very high. Tree+Δ Indexing All frequent trees size up to maxL – 1 All infrequent edges Generates graph features on the fly Query Enumerate all subtrees (maxL)

34. Experiment The number of index features used by FG-Index or SwiftIndex is much smaller than gIndex. This result indicates that more features in the index simply do not guarantee better performance.

35. Experiment The trends of all curves are consistent with those for the number of I/Os. gIndex shows the best performance in both cold and hot runs for a moderate dense dataset.

36. Experiment gCode performs the best for large query sizes with high density gIndex performs comparatively better for a larger number of labels since its pruning cost is relatively more effective

37. Results for Large Graph Database Since both SeqScan and C-Tree require prohibitive times to finish the experiments even with large buffer sizes, we exclude them from a large graph database. As for gCode, we can run experiments with a 1 GByte buffer and hot run; with smaller buffer sizes than 1 GByte and cold run, we are unable to finish the experiments within a week.

38. Results for Large Graph Database FG-Index’s pruning power is up to 13.09 times lower than gIndex, since FG-Index uses a strategy to select a subset of features in its index to minimize the filtering cost.

39. Results for Large Graph Database For Q4, FG-Index performs the best due to its verification-free strategy For Q8 ∼ Q12, gIndex performs the best since its pruning power is the best For Q16 ∼ Q24, either SwiftIndex or FG-Index performs the best since their posting list intersection costs are the least.

40. Results for Large Graph Database Although gIndex performs worse than SwiftIndex and FG-Index in the number of I/Os for large query sizes, it performs the best for all query sizes except Q4 due to a good combination of the lowest number of candidates and low disk I/O costs.

41. Conclusion Overall winner: gIndex. Large query on dense graph, we recommend gCode. Souce code: http://www.igraph.or.kr/http://www.igraph.or.kr/