The Global Packing Number of a Fat-Tree Network

Abstract

Data centers play an important role in today’s Internet development. Research to find scalable architecture and efficient routing algorithms for data center networks has gained popularity. The fat-tree architecture, which is essentially a folded version of a Clos network, has proved to be readily implementable and is scalable. In this paper, we investigate routing on a fat-tree network by deriving its global packing number and by presenting explicit algorithms for the construction of optimal, load-balanced routing solutions. Consider an optical network that employs wavelength division multiplexing in which every user node sets up a connection with every other user node. The global packing number is basically the number of wavelengths required by the network to support such a traffic load, under the restriction that each source-to-destination connection is assigned a wavelength that remains constant in the network. In mathematical terms, consider a bidirectional, simple graph, $G$ and let $N\subseteq V(G)$ be a set of nodes. A path system $\mathcal {P}$ of $G$ with respect to $N$ consists of $|N|(|N|-1)$ directed paths, one path to connect each of the source-destination node pairs in $N$ . The global packing number of a path system $\mathcal {P}$ , denoted by $\Phi (G,N,\mathcal {P})$ , is the minimum integer $k$ to guarantee the existence of a mapping $\phi :\mathcal {P}\to \{1,2,\ldots, k\}$ , such that $\phi (P)\neq \phi (\widehat {P})$ if $P$ and $\widehat {P}$ have common arc(s). The global packing number of $(G,N)$ , denoted by $\Phi (G,N)$ , is defined to be the minimum $\Phi (G,N,\mathcal {P})$ among all possible path systems $\mathcal {P}$ . In additional to wavelength division optical networks, this number also carries significance for networks employing time division multiple access. In this paper, we compute by explicit route construction the global packing number of $(\text {T}_{n},N)$ , where T n denotes the topology of the $n$ -ary fat-tree network, and $N$ is considered to be the set of all edge switches or the set of all supported hosts. We show that the constructed routes are load-balanced and require minimal link capacity at all network links.