1. Graph Algebra with Pattern Matching and Aggregation Support 1

2. Nowadays Graph Variety of Sources ◦ Scientific Studies ◦ Business Activities ◦ Social Needs ◦ Internet Data are often of ◦ Large Scale ◦ Highly Liked ◦ Schema-less 2

3. Managing Graph Data Primary Role of Database ◦ Persistent store ◦ Efficient Query RDBMS ◦ Storage Model : vertex and edge as tuples ◦ Query: Link is by join Graph Database ◦ Storage Model: graphs ◦ Query: path traversal 3

4. Why not RDBMS ? Schema Issue ◦ Every data inserted may of a different schema (Web Graph) ◦ Hard to represent semi structured info Scalability Issues ◦ ACID property VS CAP theorem Query performance ◦ Difficult to optimize intensive Joins 4

5. Graph Databases and Query Languages No Universal Languages !!! 5

6. No Universal Language Like SQL? No commonly agreed algebra Relational Algebra ? ◦ Expressive, test-of-time to be effective ◦ NOT suitable for GRAPH Graph Algebra ? ◦ Still at preliminary work 6

7. Issues with Relational Algebra (RA) Defined on Tuples or Set of Tuples ◦ Mismatch with graph nature ◦ Operators loose semantics  What is Union, Intersection, Join in GRAPH? ◦ I/O type ?  Tables not GRAPH Domain centric, not Data centric ◦ Don’t anticipate out-of-order data ◦ Treat Tuples as independent  Didn’t aware the links among Tuples  Queries written using RA are verbose and complex 7

8. Advantage of Graph Algebra An algebra itself is a query language ◦ Easy to work out a language with Strong theoretic support Evaluate expressiveness of given languages ◦ Justify when to use what: Gremlin, Cypher etc. Query Optimization ◦ Operator order EQUALS execution plan ◦ Algebraic Equivalence IMPLIES query optimization 8

9. Advantage of Graph Algebra Separation of Query and System: ◦ One can write Query on any system as long as common algebra is supported. ◦ Knowing RA, one can write SQL, PL/SQL, MS/SQL on MySQL, Oracle, SQLServer Integrate new operators to database: ◦ Current graph database systems didn’t support newly developed queries:  Graph OLAP, Graph Cube, Graph Aggregation etc. ◦ Proper Algebra can incorporate these operators 9

10. Existing Works on Graph Algebra Graph QL [1] ◦ A graph based algebra, operators are based on graphs ◦ Selection ◦ Join – not properly defined ◦ Template VAQL [2] ◦ Focused on visualization ◦ Selection ◦ Aggregation – restricted ◦ Visualization Selection is restricted on isomorphism Aggregation is not defined over edges No algebra equivalence [1] He, Huahai, and Ambuj K. Singh. "Graphs-at-a-time: query language and access methods for graph databases." Proceedings of the 2008 ACM SIGMOD international conference on Management of data. ACM, 2008. [2] Shaverdian, Anna A., et al. "A graph algebra for scalable visual analytics." Computer Graphics and Applications, IEEE 32.4 (2012): 26-33. 10

11. What we want for a Graph Algebra? Universal ◦ Independent of graph types:  Directed VS Undirected. Simple VS Hyper. Homogeneous VS heterogeneous. Expressive ◦ Able to answer typical graph queries:  Pattern match, Reachability, Path finding etc. ◦ Cover Relational Algebra (RA)  This ensures that graph database can handle relational data as well Scale ◦ Able to manage data in-scale  Support queries to summarize, aggregate data 11

12. Extended Algebra – Graph Model 12

13. Extended Algebra – Operators 13

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17. P(v1,v1) and P(v4,v5) are true 17

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19. [1] Fan, Wenfei, et al. "Adding regular expressions to graph reachability and pattern queries." Data Engineering (ICDE), 2011 IEEE 27th International Conference on. IEEE, 2011. 19

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23. Expressiveness This set of operators are more expressive than Relational Algebra and Graph QL It can represent many graph queries ◦ Reachability ◦ Graph Cube computation ◦ I-OLAP and T-OLAP 23

24. Algebra Equivalence When operators are chained up, they can form a query execution plan Find the network induced by the person whose friends comment on each other’s posts with birthday greater than 1989. Output those names as a graph friend Comment friend Base Graph Matched Result Restriction v.name V-Unification 24

25. Algebra Equivalence To generate multiple execution plans for a same query, we need theoretic support: 25

26. Conclusion Graph Algebra plays an important role in graph database development We make one step forward by proposing a Graph Algebra which: ◦ extends existing algebraic work with  Regular pattern matching  Aggregation ◦ is expressive and well-defined ◦ contains equivalence rules for further query optimization 26

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