Communication costs of Strassen's matrix multiplication

Grey Ballard, James Demmel, Olga Holtz, Oded Schwartz

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


Algorithms have historically been evaluated in terms of the number of arithmetic operations they performed. This analysis is no longer sufficient for predicting running times on today's machines. Moving data through memory hierarchies and among processors requires much more time (and energy) than performing computations. Hardware trends suggest that the relative costs of this communication will only increase. Proving lower bounds on the communication of algorithms and finding algorithms that attain these bounds are therefore fundamental goals. We show that the communication cost of an algorithm is closely related to the graph expansion properties of its corresponding computation graph. Matrix multiplication is one of the most fundamental problems in scientific computing and in parallel computing. Applying expansion analysis to Strassen's and other fast matrix multiplication algorithms, we obtain the first lower bounds on their communication costs. These bounds show that the current sequential algorithms are optimal but that previous parallel algorithms communicate more than necessary. Our new parallelization of Strassen's algorithm is communication-optimal and outperforms all previous matrix multiplication algorithms.

Original languageAmerican English
Pages (from-to)107-114
Number of pages8
JournalCommunications of the ACM
Issue number2
StatePublished - Feb 2014
Externally publishedYes


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