Parity-Checked Strassen Algorithm
share
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× 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× 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.
READ FULL TEXT