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.
Keywords
- Matrix Multiplication
- Arithmetical Operation
- Deterministic Algorithm
- Probabilistic Algorithm
- Quadratic Time
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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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