Routing Number Of A Pyramid

In this short note we give the routing number of pyramid graph under the routing via matching model introduced by Alon et al. This model can be viewed as a communication scheme on a distributed network. The nodes in the network can communicate via matchings (a step), where a node exchanges data with its partner. Formally, given a connected graph G with vertices labeled from [1,...,n] and a permutation π giving the destination of pebbles on the vertices the problem is to find a minimum step routing scheme. This is denoted as the routing time rt(G,π) of G given π. We show that a d-dimensional pyramid with m levels has a routing number of O(dN^1/d).