1. Carlos Ordonez, Predrag T. Tosic
Time Complexity and Parallel Speedup of Relational Queries to Solve Graph Problems
Carlos Ordonez, Predrag T. Tosic 1

2. Graph analytics
Exploration
Path
Connectivity
Structure

3. Our contributions Unify many algorithms into two iterations
Relational queries
Time complexity based on matrix density
Parallel processing

4. Preliminaries G definition: G=(V,E), n=|V|,m=|E|
E sparse matrix: m=O(n)
Iterative algorithms
Matrix-Vector multiplication
Matrix-Matrix multiplication

5. Basic Matrix-Vector Iteration
| S0 |=1 or |S0|=n
S= S*E
Semiring switching operators:
*/sum()
+/min()
*/min

6. Basic Matrix-Matrix R=R*E E+=E+ E2 + .. + Ek Semiring applies as well
Until fixpoint or max path length k

7. Relational Algebra

8. Time complexity merge sort
Matrix-vector: O(n log n)
Matrix-matrix:
O(m log m) if m=O(f(n)) not assumed
from O(n) to O(n3)
clique of size K=O(n)

9. Fast algorithm termination immediate fixpoint

10. Matrix-matrix: Time complexity
Tree
Complete graph

11. Hardness of Matrix-Matrix multiplication

12. Parallel processing Matrix-vector Partition S replicate S
Matrix-matrix, consider edge (i,j)
Partition by source vertex i: neighbors are local
Partition by destination vertex j: neighbors are local
Partition by edge i,j: neighbors not local

13. Parallel processing speedup
sparse :
dense matrix-vector
dense matrix-matrix
Partitioning
Hashing
Sorting: parallel merge sort
Computation
Local
Distributed

14. Conclusions Unified two families of graph algorithms
Relational algebra expressions to evaluate matrix-matrix and matrix-vector multiplication
Characterize O() and speedup assuming sparse or dense adjacency matrix