Some linear-time algorithms for systolic arrays

TL;DR

This paper reviews linear-time algorithms implemented on systolic arrays for problems like GCD computation, solving Toeplitz systems, and eigenvalue problems, highlighting their theoretical efficiency and practical applications.

Contribution

It presents new linear-time algorithms for GCD, Toeplitz systems, and eigenvalue problems on systolic arrays, demonstrating their efficiency and potential applications.

Findings

GCD of polynomials computed in O(n) time on linear arrays

Toeplitz systems solved in O(n) time with constant memory cells

Eigenvalue problem solved in O(nS(n)) time with 2D arrays

Abstract

We survey some results on linear-time algorithms for systolic arrays. In particular, we show how the greatest common divisor (GCD) of two polynomials of degree n over a finite field can be computed in time O(n) on a linear systolic array of O(n) cells; similarly for the GCD of two n-bit binary numbers. We show how n * n Toeplitz systems of linear equations can be solved in time O(n) on a linear array of O(n) cells, each of which has constant memory size (independent of n). Finally, we outline how a two-dimensional square array of O(n)* O(n) cells can be used to solve (to working accuracy) the eigenvalue problem for a symmetric real n* n matrix in time O(nS(n)). Here S(n) is a slowly growing function of n; for practical purposes S(n) can be regarded as a constant. In addition to their theoretical interest, these results have potential applications in the areas of error-correcting codes, symbolic and algebraic computations, signal processing and image processing.