Linear Complexity

Abstract

In this chapter we study the linear complexity of a matrix A, which is the minimum number of additions, subtractions, and scalar multiplications to compute the product AX, where X is a generic input vector. In Sect. 13.2 we first follow Winograd to determine the exact linear complexity of a generic matrix, and then proceed with deriving reasonable upper complexity bounds for some classes of structured matrices. There, we also prove Morgenstern’s theorem [382], which yields a lower bound of order n log n for the number of additions and multiplications by scalars of bounded absolute value to compute the discrete Fourier transform of length n over the complex field. A reformulation of the linear computational model in terms of graph theory is presented in Sect. 13.3. As a first application of this approach, we give a short proof of Tellegen’s theorem, which (in a special case) states that an invertible matrix and its transpose have the same linear complexity. Sect. 13.4 discusses the linear complexity of superregular and totally regular matrices in terms of graph theory. We prove Shoup and Smolensky’s theorem [477] that sometimes gives nonlinear lower bounds for superregular matrices, as well as Valiant’s theorem [525] that yields a lower bound for the linear complexity of totally regular n x n-matrices as a linear function of the minimal number of edges in an n-superconcentrator. We also prove Pinsker and Valiant’s theorem [418, 525] stating that n-superconcentrators with O(n) edges exist. The last section is concerned with discrete Fourier transforms for arbitrary finite groups. We present results of Baum, Beth, and Clausen that generalize the classical Cooley-Tukey FFT to wider classes of finite groups. In particular, Baum’s theorem [24] shows that every supersolvable group of order n has a DFT of linear complexity O (n log n).