Parity-Checked Strassen Algorithm

To multiply astronomic matrices using parallel workers subject to straggling, we recommend interleaving checksums with some fast matrix multiplication algorithms. Nesting the parity-checked algorithms, we weave a product code flavor protection. Two demonstrative configurations are as follows: (A) $9$ workers multiply two $2\times 2$ matrices; each worker multiplies two linear combinations of entries therein. Then the entry products sent from any $8$ workers suffice to assemble the matrix product. (B) $754$ workers multiply two $9\times 9$ matrices. With empirical frequency $99.8\%$, $729$ workers suffice, wherein $729$ is the complexity of the schoolbook algorithm. In general, we propose probability-wisely favorable configurations whose numbers of workers are close to, if not less than, the thresholds of other codes (e.g., entangled polynomial code and PolyDot code). Our proposed scheme applies recursively, respects worker locality, incurs moderate pre- and post-processes, and extends over small finite fields.

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