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Fast Sparse Matrix Multiplication

Fast Sparse Matrix Multiplication. Raphael Yuster Haifa University (Oranim) Uri Zwick Tel Aviv University ESA 2004. Matrix multiplication. j. i. =. . Matrix multiplication. Sparse Matrix Multiplication. =. . n - number of rows and columns m - number of non-zero elements.

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Fast Sparse Matrix Multiplication

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  1. Fast SparseMatrix Multiplication Raphael YusterHaifa University (Oranim) Uri ZwickTel Aviv University ESA 2004

  2. Matrix multiplication j i = 

  3. Matrix multiplication

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

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

  6. Matrix multiplication Can we do something better?

  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 Complexity = 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 0 =  0

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

  14. B1 A2 A1 B2 Complexity: The combined algorithm Fast rectangularmatrix multiplication Naïve sparsematrix multiplication Assume:a1b1 ≥ a2b2 ≥ … ≥ anbn Choose: 0 ≤ p ≤ n Compute:AB = A1B1+ A2B2

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

  16. Complexity of new algorithm: m0.64n1.46+n2+o(1) Multiplying three sparse matrices   A B C n - number of rows and columnsm - 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, algorithmfor sparse matrix multiplication? • A faster algorithm for multiplying three or more sparse matrices? • An O(m1-n1+) transitive closure algorithm?

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