Lower bounds for sorting on mesh-connected architectures

Abstract

Lower bounds for sorting on mesh-connected arrays of processors are presented. For sorting N=n1 n 2...n r elements on an n 1×n2×... ×n r array 2(n 1+...+n r−1)+n r data interchange steps are needed asymptotically. For two dimensions these bounds are asymptotically best possible provided that n 1 and n 2 are powers of 2. In this case the generalized s 2-way merge sort of Thompson and Kung turns out to be asymptotically optimal. The minimal asymptotic bound of 2 √2N interchange steps can be obtained only by sorting algorithms suitable for √N/2×√2N meshes. For r≧3 dimensions an analysis of aspect-ratios also demonstrates that there exist mesh-connected architectures which are better suited for sorting than simple r-dimensional cubes.