Faster Combinatorial Algorithms for Determinant and Pfaffian

Abstract

Computation of a determinant is a very classical problem. The related concept is a Pfaffian of a matrix defined for skew-symmetric matrices. The classical algorithm for computing the determinant is Gaussian elimination. It needs O(n 3) additions, subtractions, multiplications and divisions. 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 Mahajan-Vinay 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.