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

2. 2 Matrix multiplication  = i j

3. 3 AuthorsComplexity - n3n3 Strassen (1969) n 2.81 Coppersmith, Winograd (1990) n 2.38

4. 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. 5 Sparse Matrix Multiplication  = Each element of B is multiplied by at most n elements from A. Complexity: mn j k k

6. 6 Matrix multiplication AuthorsComplexity Coppersmith, Winograd (1990) n 2.38 - mn Can we do something better? here m 0.7 n 1.2 +n 2+o(1)

7. 7 Comparison r (m=n r )  n 2.38 mn m 0.7 n 1.2 +n 2 Complexity = n 

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

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

10. 10 Regular case: Best case for naïve algorithm

11. 11 Worst case for naïve algorithm

12. 12 Worst case for naïve algorithm  = 0 0

13. 13 Rectangular Matrix multiplication Coppersmith (1997): Complexity ≤ n 1.85 p 0.54 +n 2+o(1) For p ≤ n 0.29, complexity = n 2+o(1) !!!  = n p p n n n

14. 14 The combined algorithm Assume: a 1 b 1 ≥ a 2 b 2 ≥ … ≥ a n b n Choose: 0 ≤ p ≤ n Compute: AB = A 1 B 1 + A 2 B 2 Complexity: Fast rectangular matrix multiplication Naïve sparse matrix multiplication A1A1 A2A2 B1B1 B2B2

15. 15 Analysis of combined algorithm Theorem: There exists a 1≤p≤n for which Lemma:

16. 16 Multiplying three sparse matrices   A C B Complexity of new algorithm: m 0.64 n 1.46 +n 2+o(1) n - number of rows and columns m - number of non-zero elements

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

18. 18 Open problems A faster, more sophisticated, algorithm for sparse matrix multiplication? A faster algorithm for multiplying three or more sparse matrices? An O(m 1-  n 1+  ) transitive closure algorithm?