Now We Look at the 2-dimensional Layout of M

restricted to the dirty slices. It is almost the 2-dimensional mesh considered in Section 2. The diier-ence is that 2r + 2 leftmost and rightmost columns do not contain connections which correspond to Odd-Even Transposition Sort. Additionally, there are columns in the middle of the mesh that contain comparators implementing vertical comparisons in vertical slices. However, using Proposition 1.9, one can rewrite the proof of Section 2 to deal with such a mesh and prove that the time required for sorting dirty slices is O(q log(c 3 p)). It follows that the total time required for sorting is O(q log(pr)). For q = (n 1=3 log 2=3); p; r = (n 1=3 = log 1=3 n), we get O(n 1=3 log 5=3 n). Problem The best sorting time that we have obtained using periodic networks of constant depth is O(log 2 n). There is no known lower bound for sorting time on periodic comparator networks of constant depth other than (log n), which holds for arbitrary comparator networks. It would be interesting to close this gap. Acknowledgment We are grateful to Friedhelm Meyer auf der Heide for presenting us the problem of periodic sorting with constant depth networks. He also pointed to many ideas that nally have been implemented by our algorithms. We thank Endr e Szemer edi for remarks concerning an early version of the paper. Some parallel sorts on a mesh-connected processor array and their time eeciency. 12 in the 2-dimensional case. We assume that q and p are odd, r is even, and q = (p log r), p = (r) are appropriately chosen. The algorithmperforms parallel steps G, H and J periodically. It is convenient to consider a 2-dimensional layout of the nodes of M obtained by putting horizontal slices into a pr (q + 1) rectangle, so that the node P k;j;i is placed at the position ((k ? 1)p + j; i) (see Fig. 12). This layout and ordering of the nodes horizontal −slice 1. .. horizontal −slice 2 horizontal −slice r Figure 12: Representing M as a 2-dimensional mesh is the same as for the 2-dimensional mesh. Steps G and J correspond to steps A and C, respectively, of the algorithm on the 2-dimensional mesh. Step H is more complex and it implements all comparisons performed by the algorithm working on d-dimensional meshes on vertical slices. We consider three groups of vertical slices, …