# Randomized routing on meshes with buses

## Abstract

We give algorithms and lower bounds for the problem of routing *k*-permutations on *d*-dimensional MIMD meshes with row and column buses We prove a lower bound for routing permutations (the case *k*=1) on *d*-dimensional meshes. For *d*=2, 3 and 4 the lower bound is respectively 0.69·*n*, 0.72·*n* and 0.76·*n* steps; the bound increases monotonically with *d* and is at least (1−1/*d*)·*n* steps for all *d*≥5. Previously, a bisection argument had been used to show that for all *d*≥1, 0.66·*n* steps axe required for this problem (i.e., the lower bound did not increase with increasing *d*). These lower bounds hold for off-line routing as well. We give a general algorithm that routes *k*-permutations on *d*-dimensional meshes in min{(2−1/*d*) · *k* · *n*, max{4/3 · *d* · *n, k* · *n*/3}} + o(*d* · *k* · *n*) steps, for all *k, d*≥1. This improves considerably on previous results for many values of *k* and *d*. In particular, the routing time for permutations is bounded by 2 · *n*, for all 1≤*d*<*n*^{1/3}, and the routing time is optimal for all *k*≥4 · *d*. More specialized algorithms have better performance for routing on 2-dimensional meshes. A simple algorithm routes 2-permutations in 1.39 · *n* steps, and a more sophisticated one routes permutations in 0.78 · *n* steps. This is the first algorithm that routes permutations on the 2-dimensional mesh in less than *n* steps. The algorithms are randomized, on-line and achieve the given routing times with high probability.

## Keywords

parallel computation algorithms packet routing meshes buses lower bounds randomization coloring## Preview

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