Fast%20Sparse%20Matrix%20Multiplication

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Fast SparseMatrix Multiplication

• Raphael YusterHaifa University (Oranim)
• Uri ZwickTel Aviv University
• ESA 2004

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Matrix multiplication
j
i

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

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n - number of rows and columnsm - number of
non-zero elements
The distribution of the non-zeroelements in the
matrices is arbitrary!
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Sparse Matrix Multiplication
j
k
k

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Each element of B is multiplied by at most n
elements from A. Complexity mn
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Matrix multiplication
Authors Complexity
Coppersmith, Winograd (1990) n2.38
- mn
here m0.7n1.2n2o(1)
Can we do something better?
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Comparison
mn
m0.7n1.2n2
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n2.38
Complexity n?
r (mnr)
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A closer look at the naïve algorithm

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Complexity of the naïve algorithm
Complexity
where
Can it really be that bad?
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Best case for naïve algorithm
Regular case
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Worst case for naïve algorithm
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Worst case for naïve algorithm
0

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0
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Rectangular Matrix multiplication
Coppersmith (1997) Complexity
n1.85p0.54n2o(1)
For p n0.29, complexity n2o(1) !!!
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The combined algorithm
Fast rectangularmatrix multiplication
Naïve sparsematrix multiplication
Assume a1b1 a2b2 anbn
Choose 0 p n
Compute AB A1B1 A2B2
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Analysis of combined algorithm
Theorem There exists a 1pn for which
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Multiplying three sparse matrices
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A
B
C
n - number of rows and columnsm - number of
non-zero elements
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Applications

• Computing the square of a sparse graph
• Finding short cycles (YZ04)
• Other applications?

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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?