Routing on meshes in optimum time and with really small queues

We consider permutation routing problems on 2D and 3D mesh-connected computers with side length n. Our main result is a deterministic online algorithm routing on 2D meshes, operating in worst-case time T = 2n + 𝒪(1) and with queue size Q = 3. We also develop offline routing algorithms with performance bounds T = 2n - 1 and Q = 2 for 2D meshes, and T = 3n - 2 and Q = 4 for 3D meshes. We also show that is it possible to route most of the permutations on 2D meshes offline in time T = 2n - 2 with Q = 1.