1. Fast Sparse Matrix Multiplication
Raphael Yuster Haifa University (Oranim)
Uri Zwick Tel Aviv University
ESA 2004

2. Matrix multiplicationj i  =

3. Matrix multiplication
Authors
Complexity - n3
Strassen (1969)
n2.81
Coppersmith, Winograd (1990)
n2.38

4. Sparse Matrix Multiplication =
n - number of rows and columns m - number of non-zero elements
The distribution of the non-zero elements in the matrices is arbitrary!

5. Sparse Matrix Multiplicationj k k = 
Each element of B is multiplied by at most n elements from A.
Complexity: mn

6. Matrix multiplication
Authors
Complexity
Coppersmith, Winograd (1990)
n2.38 - mn
here
m0.7n1.2+n2+o(1)
Can we combine the two?

7. Comparison mn
m0.7n1.2+n2 
n2.38
Complexity = n
r (m=nr)

8. A closer look at the naïve algorithm =  =

9. Complexity of the naïve algorithm
where
Can it really be that bad?

10. Best case for naïve algorithm
Regular case:

11. Worst case for naïve algorithm

12. Worst case for naïve algorithm =

13. Rectangular Matrix multiplication = n p
Coppersmith (1997): Complexity ≤ n1.85p0.54+n2+o(1)
For p ≤ n0.29, complexity = n2+o(1) !!!

14. The combined algorithm
Fast rectangular matrix multiplication
Naïve sparse matrix multiplication
Assume: a1b1 ≥ a2b2 ≥ … ≥ anbn
Choose: ≤ p ≤ n
Compute: AB = A1B1+ A2B2
Complexity:

15. Analysis of combined algorithm
Theorem: There exists a p for which
Lemma:

16. Multiplying three sparse matrices  B C
Complexity of new algorithm:
m0.64n1.46+n2+o(1)
n - number of rows and columns m - number of non-zero elements

17. Applications Computing the square of a sparse graph
Finding short cycles (YZ’04)
Other applications?

18. Open problems
A faster, more sophisticated, algorithm for sparse matrix multiplication?
A faster algorithm for multiplying three or more sparse matrices?
An O(m1-n1+) transitive closure algorithm?