The I/O complexity of hybrid algorithms for square matrix multiplication
Abstract
Asymptotically tight lower bounds are derived for the I/O complexity of a general class of hybrid algorithms computing the product of $n \times n$ square matrices combining ``\emph{Strassenlike}'' fast matrix multiplication approach with computational complexity $\Theta{n^{\log_2 7}}$, and ``\emph{standard}'' matrix multiplication algorithms with computational complexity $\Omega\left(n^3\right)$. We present a novel and tight $\Omega\left(\left(\frac{n}{\max\{\sqrt{M},n_0\}}\right)^{\log_2 7}\left(\max\{1,\frac{n_0}{M}\}\right)^3M\right)$ lower bound for the I/O complexity a class of ``\emph{uniform, nonstationary}'' hybrid algorithms when executed in a twolevel storage hierarchy with $M$ words of fast memory, where $n_0$ denotes the threshold size of subproblems which are computed using standard algorithms with algebraic complexity $\Omega\left(n^3\right)$. The lower bound is actually derived for the more general class of ``\emph{nonuniform, nonstationary}'' hybrid algorithms which allow recursive calls to have a different structure, even when they refer to the multiplication of matrices of the same size and in the same recursive level, although the quantitative expressions become more involved. Our results are the first I/O lower bounds for these classes of hybrid algorithms. All presented lower bounds apply even if the recomputation of partial results is allowed and are asymptotically tight. The proof technique combines the analysis of the Grigoriev's flow of the matrix multiplication function, combinatorial properties of the encoding functions used by fast Strassenlike algorithms, and an application of the LoomisWhitney geometric theorem for the analysis of standard matrix multiplication algorithms. Extensions of the lower bounds for a parallel model with $P$ processors are also discussed.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.12804
 Bibcode:
 2019arXiv190412804D
 Keywords:

 Computer Science  Data Structures and Algorithms