Block gossiping on grids and tori: Deterministic sorting and routing match the bisection bound

Abstract

Deterministic sorting and routing on r-dimensional n × … × n grids of processors is studied. For h - h problems, h ≥ 4r, where each processor initially and finally contains at most h elements, we show that the general h - h sorting as well as h - h routing problem can be solved within hn/2 + o(hr2n). That is, the bisection bound is asymptotically tight for deterministic h - h sorting and h - h routing. On an r-dimensional torus, a grid with wrap-arounds, the number of transfer steps is hn/4 + o(hrn), again matching the corresponding bisection bound. This shows that inspire of the fact that routing problems contain more information at the beginning than the sorting problems there is no substantial difference between them on grids and tori: The results are possible by a new method where subsets of packets and information are uniformly distributed to the whole grid.