Optimal Deterministic Sorting and Routing on Grids and Tori with Diagonals 1

We present deterministic sorting and routing algorithms for grids and tori with additional diagonal connections. For large loads (h 12), where each processor has at most h data packets in the beginning and in the end, the sorting problem can be solved in optimal hn=6 + o(n) and hn=12 + o(n) steps for grids and tori with diagonals, respectively. For smaller loads, we present a new concentration technique that yields very fast algorithms for h < 12. For a load of 1, the previously most studied case, sorting only takes 1:2n + o(n) steps and routing only 1:1n + o(n) steps. For tori, we can present optimal algorithms for all loads h 1. The above algorithms all use a constant size memory for all processors and never copy or split packets, a property that the corresponding lower bounds make use of. If packets may be copied, 1{1 sorting can be done in only 2n=3 + o(n) on a torus with diagonals. Generally gaining a speedup of 3 by only doubling the number of communication links compared to a grid without diagonals, our work suggests building grids and tori with diagonals.