Communication complexity of the Gaussian elimination algorithm on multiprocessors

This paper proposes a few lower bounds for communication complexity of the Gaussian elimination algorithm on multiprocessors. Three types of architectures are considered: a bus architecture, a nearest neighbor ring network, and a nearest neighbor grid network. It is shown that for the bus and the ring architectures, the minimum communication time is O(N²), independent of the number of processors, while for the grid it is reduced to O(k ^{built-1 2}N²)+O(k ^{built1 2}N) for a lock step Gaussian elimination algorithm, and to O(k ^{built-1 2}N²)+O(k ^{built1 2}) for any pipelined Gaussian elimination algorithm, where k is the total number of processors. The practical implications of these bounds are discussed.