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

• Published in: IEEE transactions on parallel and distributed systems Vol. 9; no. 8; pp. 705 - 720
• Main Authors: Li, Keqin, Pan, Yi, Zheng, Si Qing
• Format: Journal Article
• Language: English
• Published: IEEE 01.08.1998
• Subjects: Concurrent computing; Costs; Eigenvalues and eigenfunctions; Graph theory; Optical arrays; Parallel algorithms; Parallel processing; Polynomials; Power engineering and energy; Tree graphs
• ISSN: 1045-9219; 1558-2183
• DOI: 10.1109/71.706044

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, MM/sub 1/, MM/sub 2/, MM/sub 3/, and compound algorithms C/sub 1/(/spl epsiv/)and C/sub 2/(/spl delta/). Algorithm C/sub 1/(/spl epsiv/) has adjustable time complexity in sublinear level. Algorithm C/sub 2/(/spl delta/) implies that it is feasible to achieve sublogarithmic time using /spl sigma/(N/sup 3/) processors for matrix multiplication on a realistic system. Algorithms MM/sub 3/, C/sub 1/(/spl epsiv/), and C/sub 2/(/spl delta/) all have o(/spl Nscr//sup 3/) cost and, hence, are very processor efficient. Algorithms MM/sub 1/, MM/sub 3/, and C/sub 1/(/spl epsiv/) are general-purpose matrix multiplication algorithms, where the array elements are in any ring. Algorithms MM/sub 2/ and C/sub 2/(/spl delta/) 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/sub 2/ and C/sub 2/(/spl delta/) to unbounded integers and reals are also discussed.