Towards practical permutation routing on meshes

We consider the permutation routing problem on two-dimensional n×n meshes. To be practical, a routing algorithm is required to ensure very small queue sizes Q, and very low running time T, not only asymptotically but particularly also for the practically important n up to 1000. With a technique inspired by a scheme of Kaklamanis/Krizanc/Rao, we obtain a near-optimal result: T=2·n+𝒪(1) with Q=2. Although Q is very attractive now, the lower order terms in T make this algorithm highly impractical. Therefore we present simple schemes which are asymptotically slower, but have T≃3·n for all n and Q between 2 and 8