# The I/O complexity of Strassen's matrix multiplication with recomputation

arXiv: https://arxiv.org/abs/1605.02224 · HTML: https://mlsanity.com/paper/1605.02224/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2016.

## Built on by (most cited first, 1 of 1)

- [The I/O Complexity of Attention, or How Optimal is Flash Attention?](https://mlsanity.com/paper/2402.07443/) (2024)

## Abstract (arXiv)

A tight Omega((n/√M)log2 7M) lower bound is derived on the io complexity of Strassen's algorithm to multiply two n × n matrices, in a two-level storage hierarchy with M words of fast memory. A proof technique is introduced, which exploits the Grigoriev's flow of the matrix multiplication function as well as some combinatorial properties of the Strassen computational directed acyclic graph (CDAG). Applications to parallel computation are also developed. The result generalizes a similar bound previously obtained under the constraint of no-recomputation, that is, that intermediate results cannot be computed more than once. For this restricted case, another lower bound technique is presented, which leads to a simpler analysis of the io complexity of Strassen's algorithm and can be readily extended to other "Strassen-like" algorithms.