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Correcting matrix products over the ring of integers (2307.12513v2)
Published 24 Jul 2023 in cs.DS and cs.DM
Abstract: Let $A$, $B$, and $C$ be three $n\times n$ matrices. We investigate the problem of verifying whether $AB=C$ over the ring of integers and finding the correct product $AB$. Given that $C$ is different from $AB$ by at most $k$ entries, we propose an algorithm that uses $O(\sqrt{k}n2+k2n)$ operations. Let $\alpha$ be the largest absolute value of an entry in $A$, $B$, and $C$. The integers involved in the computation are of $O(n3\alpha2)$.
- Combinatorial Group Testing and Its Applications. WORLD SCIENTIFIC, 2nd edition, 1999.
- Faster matrix multiplication via asymmetric hashing. In 64th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2023), Santa Cruz, CA, USA, pages 2129–2138, 2023.
- Rusins Freivalds. Probabilistic machines can use less running time. In Information Processing, Proceedings of the 7th IFIP Congress, pages 839–842. North-Holland, 1977.
- Efficiently correcting matrix products. Algorithmica, 79(2):428–443, 2017.
- Deterministic verification of integer matrix multiplication in quadratic time. In 40th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2014), Nový Smokovec, Slovakia, pages 375–382, 2014.
- Marvin Künnemann. On nondeterministic derandomization of Freivalds’ algorithm: Consequences, avenues and algorithmic progress. In 26th Annual European Symposium on Algorithms (ESA 2018), Helsinki, Finland, volume 112 of LIPIcs, pages 56:1–56:16, 2018.
- Konstantin Kutzkov. Deterministic algorithms for skewed matrix products. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013), Dagstuhl, Germany, volume 20 of LIPIcs, pages 466–477, 2013.
- Rasmus Pagh. Compressed matrix multiplication. ACM Trans. Comput. Theory, 5(3):1–17, 2013.
- Volker Strassen. Gaussian elimination is not optimal. Numer. Math., 13(4):354–356, 1969.