Optimal Deterministic Sorting and Routing on Grids and Tori with Diagonals

Abstract.

We present deterministic sorting and routing algorithms for grids and tori with additional diagonal connections. For large loads ( $h\geq 12$ ), 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 $h\geq 1$ . 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 with a grid without diagonals, our work suggests building grids and tori with diagonals.