Abstract
Let A, B and C be n×n matrices of integer numbers. We show that there is a deterministic algorithm of quadratic time complexity (w.r.t. the number of arithmetical operations) verifying whether AB=C. For the integer matrices this result improves upon the best known result by Freivalds from 1977 that only holds for a randomized (Monte Carlo) algorithm. As a consequence, we design a quadratic time nondeterministic integer and rational matrix multiplication algorithm whose time complexity cannot be further improved. This indicates that any technique for proving a super-quadratic lower bound for deterministic matrix multiplication must exploit methods which would not work for the non-deterministic case.
This research was partially supported by RVO 67985807 and the GA ČR grant No. P202/10/1333. The paper is based on the joint research of both authors which started shortly before the untimely death of Ivan Korec in 1998.
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Korec, I., Wiedermann, J. (2014). Deterministic Verification of Integer Matrix Multiplication in Quadratic Time. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds) SOFSEM 2014: Theory and Practice of Computer Science. SOFSEM 2014. Lecture Notes in Computer Science, vol 8327. Springer, Cham. https://doi.org/10.1007/978-3-319-04298-5_33
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DOI: https://doi.org/10.1007/978-3-319-04298-5_33
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