Optimal randomized algorithms for multipacket and cut through routing on the mesh

In this paper, we present a randomized algorithm for the multipacket (i.e., k − k) routing problem on an n × n mesh. The algorithm completes with high probability in at the most kn + O(k logn) parallel communication steps, with a constant queue size of O(k). The previous best known algorithm [3] takes 5 4kn + O( kn f(n) ) steps with a queue size of O(k f(n)) (for any 1 ≤ f(n) ≤ n). We will also present a randomized algorithm for the cut through with partial cuts model permutation routing problem for the mesh that completes in at the most kn + O(k logn) steps, with a constant queue size of O(k), where k is the number of flits that each packet is divided into. The previous best result [6] was also randomized and had a time bound of kn+O( kn f(n) ) with a queue size of O(k f(n)) (for any 1 ≤ f(n) ≤ n). The two algorithms that we will present are optimal with respect to queue size. The time bounds are within a factor of two of the only known lower bound.