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Routing Number Of A Pyramid

In this short note we give the routing number of pyramid graph under the \textit{routing via matching} model introduced by Alon et al\cite{5}. 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 $O(dN^{1/d})$.