Fast and Scalable Parallel Algorithms for Matrix Chain Product and Matrix Powers on Optical Buses

Abstract

Given N matrices A ₁, A 2 ..., A _N of size N − _N, the matrix chain product problem is to compute A ₁ × A ₂ × ... × A _N Given an N x N matrix A, the matrix powers problem is to calculate the first N powers of A, i.e., A, A ², A ³,... A ^N. We show that the two problems can be solved in

$$ O\left( {\frac{{N^{\alpha + 1} }} {p} + \frac{{N^{2(1 + 1/\alpha )} }} {{p^{2/\alpha } }}\log \frac{p} {N} + (\log N)^2 } \right)$$

and

$$ O\left( {\frac{{N^{\alpha + 1} }} {p} + \frac{{N^{2(1 + 1/\alpha )} }} {{p^{2/\alpha } }}\log p + \log N\log p} \right)$$

times respectively, where α < 2.3755, and p, the number of processors, can be arbitrarily chosen in the interval [1‥ N ^α+1] Our highly scalable algorithms can be implemented on a linear array with a reconfigurable pipelined bus system, which is a distributed memory system using optical interconnections.