Berkowitz's algorithm and clow sequences

Abstract

A combinatorial interpretation of Berkowitz's algorithm is presented. Berkowitz's algorithm is the fastest known parallel algorithm for computing the characteristic polynomial of a matrix. The combinatorial interpretation is based on loop covers introduced by Valiant, and sequences, defined by Mahajan and Vinay. Clow sequences turn out to capture very succinctly the computations performed by Berkowitz's algorithm, which otherwise are quite difficult to analyze. The main contribution of this paper is a proof of correctness of Berkowitz's algorithm in terms of clow sequences. multiple of log 2 n. There are two other fast parallel algorithms for computing the coefficients of the characteristic polynomial of a matrix: Chistov's algorithm and Csanky's algorithm. Chistov's algorithm is more difficult to formalize, and Csanky's algorithm works only for fields of char 0; see (8, section 13.4) for details about these two algorithms. The author's original motivation for studying Berkowitz's algorithm was the proof complexity of linear algebra. Proof Complexity deals with the complexity of formal mathematical derivations, and it has applications in lower bounds and automated theorem proving. In particular, the author was interested in the complexity of deriva- tions of matrix identities such as AB = I → BA = I (right matrix inverses are left inverses). These identities have been proposed by Cook as candidates for separating the Frege and Extended Frege proof systems. Proving (or disproving) this separation is one of the outstanding problems in propositional proof complexity; see (3) for a comprehensive exposition of this area of research. Thus, we were interested in an algorithm that could compute inverses of matrices, of the lowest complexity possible. Berkowitz's algorithm is ideal for our purposes for several reasons: • as was mentioned above, it has the lowest known complexity for computing