Routing Number Of A Pyramid

TL;DR

This paper determines the routing number of a pyramid graph under the routing via matching model, showing it scales as O(dN^{1/d}) for a d-dimensional pyramid with N nodes.

Contribution

It provides the first explicit routing number for pyramid graphs in the routing via matching model, extending understanding of communication complexity in such networks.

Findings

Routing number of d-dimensional pyramid is O(dN^{1/d})

The result applies to the routing via matching model

Improves understanding of communication schemes in pyramid networks

Abstract

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 $\pi$ 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 $rt(G,\pi)$ of $G$ given $\pi$. We show that a $d$-dimensional pyramid with $m$ levels has a routing number of $O(dN^{1/d})$.