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Improved time and space bounds for Boolean matrix multiplication

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Summary

Using modular arithmetic we obtain the following improved bounds on the time and space complexities for n × n Boolean matrix multiplication: O(n log 2 7 lognlogloglognloglogloglogn) bit operations and O(n 2loglog n) bits of storage on a logarithmic cost RAM having no multiply or divide instruction; O(n log 2 7(logn)2−1/2log 2 7(loglog n)1/2log 2 7−1) bit operations and O(n 2log n) bits of storage on a RAM which can use indirect addressing for table lookups. The first algorithm can be realized as a Boolean circuit with O(n log 2 7lognlogloglognloglogloglogn) gates. Whenever n×n arithmetic matrix multiplication can be performed in less than O(n log 2 7) arithmetic operations, our results have corresponding improvements.

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This work was supported in part by the Office of Naval Research under contract N00014-67-0204-0063, by the National Research Council of Canada under grant A4307, and by the National Science Foundation under grants MCS76-17321 and GJ-43332

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Adleman, L., Booth, K.S., Preparata, F.P. et al. Improved time and space bounds for Boolean matrix multiplication. Acta Informatica 11, 61–70 (1978). https://doi.org/10.1007/BF00264600

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