1. Unified Algorithm to Solve Several Graph Problems with Relational Queries
Wellington Cabrera, Carlos Ordonez (presenter)
University of Houston, USA

2. Motivation
Graph problems are among the most challenging problems in big data analytics (social networks, WWW, transportation networks).
Are specialized graph systems (e.g. Giraph) required to analyze big graphs?
Lot of data stored in relational databases.
Query processing: studied for a long time.

3. Definitions Let G=(V,E) , E is stored in a relational table E(i,j,v)
Table E corresponds to the adjacency matrix E, omitting zeroes
weights/distances are represented by v
If |E|= O(n), we say that E is sparse
Let S be a vector of n graph vertices, stored on table S(j,v):
v: distance, reachability, order, probability
We omit v values with no information (like inifinity for distances, 0 for probabilities)

4. Example: Directed Graph2 3 6 2 2 1 3 2 1 2 4 3 2 3 1 7 5 3

5. Bellman-Ford Reachability Topological Sort Page Rank
Graph Algorithms
Bellman-Ford
Reachability
Topological Sort
Page Rank
Main idea:
These algorithms
can be expressed as
a sequence of
vector-matrix
multiplications
How can they work in a relational database?

6. Graph algorithms over a semi-ring:

7. Algorithm Pattern:

8. Example: Vector-Matrix Multiplication with SQL queries
Vector-Matrix Multiplication (+ ,* ) semiring
SELECT S.j, sum(S.v * E.v)
FROM Sd-1 as S join E on S.j=E.i
GROUP BY j
Vector-Matrix Multiplication (min, +) semiring
SELECT S.j, min(S.v + E.v)
FROM Sd-1 as S JOIN E on S.j=E.i
In general
SELECT S.j, g()(S.v ⊕ E.v)

9. Bellman Ford Input: Table E
Output: Table Sd ( Vector with shortest distances from a source)

10. Reachability
Input: Table E
Output: Table Rd

11. PageRank
Input: Table E Output: Table Sd ( Vector with shortest distances from a source)

12. Topological Sort Input: Table E
Output: Table Sd ( Vector with shortest distances from a source)

13. Comparison of 4 algorithms:

14. Unified Algorithm Input: E, S0, R0, f(), g(), ⨂, ε, unionFlag
Optional Input: s
Output: Rd

15. Conclusions
Graph algorithms are expressed as an iteration of SPJA queries
External algorithms
Not limited by RAM
Strengths
Sparse storage
Early termination, when possible
Lightweight relational queries
Unified Algorithm
Solves 4 important and diverse graph problems
Future work: more graph algorithms

16. References
C. Ordonez, W. Cabrera, and A. Gurram. Comparing columnar, row and array DBMSs to process recursive queries on graphs, Information Systems journal, 2016 (accepted).