Matrix multiplication via arithmetic progressions
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Rectangular matrix multiplication revisited
Journal of Complexity
Fast parallel algorithms for graph matching problems
Fast parallel algorithms for graph matching problems
A combinatorial algorithm for the determinant
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
A combinatorial algorithm for Pfaffians
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
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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 pseudopolynomial 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 Õ(n3.03) using the fast matrix multiplication algorithm.