A sparse-mesh, which has PUs on the diagonal of a two-dimensional grid only, is a cost effective distributed memory machine. Variants of this machine have been considered before, but none are as simple and pure as a sparse-mesh. Various fundamental problems (routing, sorting, list ranking) are analyzed, proving that sparse-meshes have great potential. It is shown that on a two-dimensional n/spl times sparse-mesh, which has n PUs, for h=/spl omega/(n/sup /spl epsiv///spl middot/log n), h-relations can be routed in (h+o(h))//spl epsiv/ steps. The results are extended for higher dimensional sparse-meshes. On a d-dimensional n x/spl middot//spl middot//spl middot/x n sparse-mesh, with h=/spl omega/(n/sup /spl epsiv///spl middot/log n), h-relations are routed in (6/spl middot/(d-1)//spl epsiv/-4)/spl middot/(h+o(h)) steps.
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