Index Transformation Algorithms in a Linear Algebra Framework

Abstract

We present a linear algebraic formulation for a class of index transformations such as Gray code encoding and decoding, matrix transpose, bit reversal, vector reversal, shuffles, and other index or dimension permutations. This formulation unifies, simplifies, and can be used to derive algorithms for hypercube multiprocessors. We show how all the widely known properties of Gray codes, and some not so well-known properties as well, can be derived using this framework. Using this framework, we relate hypercube communications algorithms to Gauss-Jordan elimination on a matrix of 0's and 1's.