The Time Complexity of Fully Sparse Matrix Multiplication
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What is the time complexity of matrix multiplication of sparse integer matrices with m_in nonzeros in the input and m_out nonzeros in the output? This paper provides improved upper bounds for this question for almost any choice of m_in vs. m_out, and provides evidence that these new bounds might be optimal up to further progress on fast matrix multiplication. Our main contribution is a new algorithm that reduces sparse matrix multiplication to dense (but smaller) rectangular matrix multiplication. Our running time thus depends on the optimal exponent ω(a,b,c) of multiplying dense n^a× n^b by n^b× n^c matrices. We discover that when m_out=Θ(m_in^r) the time complexity of sparse matrix multiplication is O(m_in^σ+ϵ), for all ϵ > 0, where σ is the solution to the equation ω(σ-1,2-σ,1+r-σ)=σ. No matter what ω(·,·,·) turns out to be, and for all r∈(0,2), the new bound beats the state of the art, and we provide evidence that it is optimal based on the complexity of the all-edge triangle problem. In particular, in terms of the input plus output size m = m_in + m_out our algorithm runs in time O(m^1.3459). Even for Boolean matrices, this improves over the previous m^2ω/ω+1+ϵ=O(m^1.4071) bound [Amossen, Pagh; 2009], which was a natural barrier since it coincides with the longstanding bound of all-edge triangle in sparse graphs [Alon, Yuster, Zwick; 1994]. We find it interesting that matrix multiplication can be solved faster than triangle detection in this natural setting. In fact, we establish an equivalence to a special case of the all-edge triangle problem.
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