## AbstractPermutation routing is used as one of the standard tests of routing algorithms. In the permutation routing problem, each processor is the origin of at most one packet and the destination of no more than one packet. The goalis to minimize the number of time steps required to route all packets to their respective destinations. Wireless mesh networks are based on plane tessellations that divide the area into cells and give rise to triangular, square, and hexagonal grids. In this paper we study permutation routing algorithms that work on finite convex subgraphs of basic grids, under the store-and-forward $\Delta$-port model. We consider algorithms implemented independently at each node, without assuming any global knowledge about the network. I.e., distributed algorithms. We describe optimal distributed permutation routing algorithms for subgraphs of triangular and square grids that need $\ell_{max}$ (the maximum over the length of the shortest path of all packets) routing steps, and show that there is no such algorithm on the hexagonal grids. Furthermore, we show that these algorithms are oblivious and translation invariant.
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