The Global Packing Number of a Fat-Tree Network

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 <italic>global packing number</italic> 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, <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> and let <inline-formula> <tex-math notation="LaTeX">$N\subseteq V(G)$ </tex-math></inline-formula> be a set of nodes. A path system <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> with respect to <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> consists of <inline-formula> <tex-math notation="LaTeX">$|N|(|N|-1)$ </tex-math></inline-formula> directed paths, one path to connect each of the source-destination node pairs in <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>. The global packing number of a path system <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\Phi (G,N,\mathcal {P})$ </tex-math></inline-formula>, is the minimum integer <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> to guarantee the existence of a mapping <inline-formula> <tex-math notation="LaTeX">$\phi :\mathcal {P}\to \{1,2,\ldots, k\}$ </tex-math></inline-formula>, such that <inline-formula> <tex-math notation="LaTeX">$\phi (P)\neq \phi (\widehat {P})$ </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\widehat {P}$ </tex-math></inline-formula> have common arc(s). The global packing number of <inline-formula> <tex-math notation="LaTeX">$(G,N)$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\Phi (G,N)$ </tex-math></inline-formula>, is defined to be the minimum <inline-formula> <tex-math notation="LaTeX">$\Phi (G,N,\mathcal {P})$ </tex-math></inline-formula> among all possible path systems <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$(\text {T}_{n},N)$ </tex-math></inline-formula>, where T_{<italic>n</italic>} denotes the topology of the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-ary fat-tree network, and <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> 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.