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.