Fast and processor efficient parallel matrix multiplication algorithms on a linear array with a reconfigurable pipelined bus system

We present efficient parallel matrix multiplication algorithms for linear arrays with reconfigurable pipelined bus systems (LARPBS). Such systems are able to support a large volume of parallel communication of various patterns in constant time. An LARPBS can also be reconfigured into many independent subsystems and, thus, is able to support parallel implementations of divide-and-conquer computations like Strassen´s algorithm. The main contributions of the paper are as follows. We develop five matrix multiplication algorithms with varying degrees of parallelism on the LARPBS computing model; namely, MM₁, MM ₂, MM₃, and compound algorithms C₁(ε)and C₂(δ). Algorithm C₁(ε) has adjustable time complexity in sublinear level. Algorithm C₂(δ) implies that it is feasible to achieve sublogarithmic time using σ(N³) processors for matrix multiplication on a realistic system. Algorithms MM₃, C₁(ε), and C₂(δ) all have o(𝒩³) cost and, hence, are very processor efficient. Algorithms MM₁, MM₃, and C₁(ε) are general-purpose matrix multiplication algorithms, where the array elements are in any ring. Algorithms MM₂ and C₂(δ) are applicable to array elements that are integers of bounded magnitude, or floating-point values of bounded precision and magnitude, or Boolean values. Extension of algorithms MM ₂ and C₂(δ) to unbounded integers and reals are also discussed