Square meshes are not always optimal

Abstract

Mesh-connected computers with multiple buses providing broadcast facilities along rows and columns are discussed. A tight bound of Theta (n/sup 1/8/) is established for the number of rounds required for semigroup computations on n values distributed on a two-dimensional rectangular mesh of size n with a bus on every row and column. The upper bound is obtained for a skewed rectangular mesh of dimensions n/sup 3/8/*n/sup 5/8/. This result is compared to the tight bound of Theta (n/sup 1/6/) for the same problem on the square (n/sup 1/2/*n/sup 1/2/) mesh. It is shown that in the presence of multiple buses, a skewed configuration may perform better than a square configuration for certain computational tasks. The result can be extended to the d-dimensional mesh, giving a lower bound of Omega (n/sup 1/d alpha /) and an upper bound of O(d2/sup d+1/ n/sup 1/d alpha /), where alpha =2/sup d/; these bounds are optimal within constant factors for any constant d. It is noted that for d>3, the results of are mostly of theoretical interest. >