Abstract
The algorithms of Mahajan and Vinay compute determinant and Pfaffian in a completely non-classical and combinatorial way, by reducing these problems to summation of paths in some acyclic graphs. The attractive feature of these algorithms is that they are division-free. We present a novel algebraic view of these algorithms: a relation to a pseudo-polynomial dynamic-programming algorithm for the knapsack problem. The main phase of MV-algorithm can be interpreted as a computation of an algebraic version of the knapsack problem, which is an alternative to the graph-theoretic approach used in the original algorithm. Our main results show how to implement Mahajan-Vinay algorithms without divisions, in time \(\tilde{O}(n^{3.03})\) using the fast matrix multiplication algorithm.
Supported by the grant of the Polish Ministery of Science and Higher Education N 206 004 32/0806.
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Urbańska, A. (2007). Faster Combinatorial Algorithms for Determinant and Pfaffian. In: Tokuyama, T. (eds) Algorithms and Computation. ISAAC 2007. Lecture Notes in Computer Science, vol 4835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77120-3_52
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DOI: https://doi.org/10.1007/978-3-540-77120-3_52
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