k-k Routing, k-k Sorting, and Cut Through Routing on the Mesh

In this paper we present randomized algorithms for k-k routing, k-k sorting, and cut through routing. The stated resource bounds hold with high probability. The algorithm for k-k routing runs in [k/2]n+o(kn) steps. We also show that k-k sorting can be accomplished within [k/2] n+n+o(kn) steps, and cut through routing can be done in [3/4]kn+[3/2]n+o(kn) steps. The best known time bounds (prior to this paper) for all these three problems were kn+o(kn). [kn/2] is a known lower bound for all the three problems (which is the bisection bound), and hence our algorithms are very nearly optimal. All the above mentioned algorithms have optimal queue length, namely k+o(k). These algorithms also extend to higher dimensional meshes.