Randomized routing on meshes with buses

Abstract

We give algorithms and lower bounds for the problem of routing k-permututions 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 respectivdy 0.69 · n, 0.72 · n and 0.76 · 9 steps; the bound increases monotonically with d and is at least (1 - 1/d) · n steps for all d ≥ 5. Previously, u bisection argument had been used to show that for all d ≥ 1, 0.66 · n steps are 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 Ir and d. In pastlculaz, the routing time for permutations is bounded by 2 · 9n, for all 1 ≤ d