Stellar datacenter networks are a recent generic construction designed to transform a base-graph into a dual-port, server-centric datacenter network. We prove that the S-bisection width of any stellar datacenter network can be obtained from the solution of isoperimetric problems on the base-graph, provided that the base-graph is regular. We extend previous research on the stellar datacenter networks GQ⁎, instantiated with generalized hypercubes, and show that with respect to S-bisection width, GQ⁎ performs well in comparison with the dual-port datacenter network FiConn. Our work develops a strong combinatorial link between graph bisection width and throughput metrics for stellar datacenter networks.

This paper presents INRFlow, a mature, frugal, flow-level simulation framework for modelling large-scale networks and computing systems. INRFlow is designed to carry out performance-related studies of interconnection networks for both high performance computing systems and datacentres. It features a completely modular design in which adding new topologies, routings or traffic models requires minimum effort. Moreover, INRFlow includes two different simulation engines: a static engine that is able to scale to tens of millions of nodes and a dynamic one that captures temporal and causal relationships to provide more realistic simulations. We will describe the main aspects of the simulator, including system models, traffic models and the large variety of topologies and routings implemented so far. We conclude the paper with a case study that analyses the scalability of several typical topologies. INRFlow has been used to conduct a variety of studies including evaluation of novel topologies and routings (both in the context of graph theory and optimization), analysis of storage and bandwidth allocation strategies and understanding of interferences between application and storage traffic. • We present our flow-level simulation framework INRFlow. • It is a mature, flexible and efficient tool for simulating large scale systems. • It models network, storage, scheduler and applications. • It has been used extensively for our research in the past. • INRFlow is open source and programmed in C.

We undertake the first formal analysis of the role of symmetry, interpreted broadly, in the design of server-centric datacenter networks. Although symmetry has been mentioned by other researchers, we explicitly relate it to various specific, structural, graph-theoretic properties of datacenter networks. Our analysis of symmetry is motivated by the need to ascertain the usefulness of a datacenter network as regards the support of network virtualization and prevalent communication patterns in multitenanted clouds. We argue that a number of structural concepts relating to symmetry from general interconnection networks, such as recursive-definability, the existence and dynamic construction of spanning trees, pancyclicity, and variations in Hamiltonicity, are appropriate topological metrics to use in this regard. In relation to symmetry, we highlight the relevance of algebraic properties and algebraic constructions within datacenter network design. Built upon our analysis of symmetry, we outline the first technique to embed guest datacenter networks in a host datacenter network that is specifically oriented towards server-centric datacenter networks. In short, we provide the graph-theoretic foundations for the design of server-centric datacenter networks so as to support network virtualization and communication patterns in cloud computing.

The first dual-port server-centric datacenter network, FiConn, was introduced in 2009 and there are several others now in existence; however, the pool of topologies to choose from remains small. We propose a new generic construction, the stellar transformation, that dramatically increases the size of this pool by facilitating the transformation of well-studied topologies from interconnection networks, along with their networking properties and routing algorithms, into viable dual-port server-centric datacenter network topologies. We demonstrate that under our transformation, numerous interconnection networks yield datacenter network topologies with potentially good, and easily computable, baseline properties. We instantiate our construction so as to apply it to generalized hypercubes and obtain the datacenter networks GQ⋆. Our construction automatically yields routing algorithms for GQ⋆ and we empirically compare GQ⋆ (and its routing algorithms) with the established datacenter networks FiConn and DPillar (and their routing algorithms); this comparison is with respect to network throughput, latency, load balancing, fault-tolerance, and cost to build, and is with regard to all-to-all, many all-to-all, butterfly, random, hot-region, and hot-spot traffic patterns. We find that GQ⋆ outperforms both FiConn and DPillar (sometimes significantly so) and that there is substantial scope for our stellar transformation to yield new dual-port server-centric datacenter networks that are a considerable improvement on existing ones.

We present significantly improved one-to-one routing algorithms in the datacenter networks HCN and BCN in that our routing algorithms result in much shorter paths when compared with existing routing algorithms. We also present a much tighter analysis of HCN and BCN by observing that there is a very close relationship between the datacenter networks HCN and the interconnection networks known as WK-recursive networks. We use existing results concerning WK-recursive networks to prove the optimality of our new routing algorithm for HCN and also to significantly aid the implementation of our routing algorithms in both HCN and BCN. Furthermore, we empirically evaluate our new routing algorithms for BCN, against existing ones, across a range of metrics relating to path-length, throughput, and latency for the traffic patterns all-to-one, bisection, butterfly, hot-region, many-all-to-all, and uniform-random, and we also study the completion times of workloads relating to MapReduce, stencil and sweep, and unstructured applications. Not only do our results significantly improve routing in our datacenter networks for all of the different scenarios considered but they also emphasize that existing theoretical research can impact upon modern computational platforms.

DPillar has recently been proposed as a server-centric datacenter network and is combinatorially related to (but distinct from) the well-known wrapped butterfly network. We explain the relationship between DPillar and the wrapped butterfly network before proving that the underlying graph of DPillar is a Cayley graph; hence, the datacenter network DPillar is node-symmetric. We use this symmetry property to establish a single-path routing algorithm for DPillar that computes a shortest path and has time complexity O(k), where k parameterizes the dimension of DPillar (we refer to the number of ports in its switches as n). Our analysis also enables us to calculate the diameter of DPillar exactly. Moreover, our algorithm is trivial to implement, being essentially a conditional clause of numeric tests, and improves significantly upon a routing algorithm earlier employed for DPillar. Furthermore, we provide empirical data in order to demonstrate this improvement. In particular, we empirically show that our routing algorithm improves the average length of paths found, the aggregate bottleneck throughput, and the communication latency. A secondary, yet important, effect of our work is that it emphasises that datacenter networks are amenable to a closer combinatorial scrutiny that can significantly improve their computational efficiency and performance.

DCell has been proposed for data centers as a server-centric interconnection network structure. DCell can support millions of servers with high network capacity by only using commodity switches. With one exception, we prove that a |$k$| level DCell built with |$n$| port switches is Hamiltonian-connected for |$k \geq 0$| and |$n \geq 2$|⁠. Our proof extends to all Generalized DCell connection rules for |$n\ge 3$|⁠. Then, we propose an |$O(t_k)$| algorithm for finding a Hamiltonian path in |$DCell_{k}$|⁠, where |$t_k$| is the number of servers in |$DCell_{k}$|⁠. Furthermore, we prove that |$DCell_{k}$| is |$(n +k - 4)$|-fault Hamiltonian-connected and |$(n +k - 3)$|-fault Hamiltonian. In addition, we show that a partial DCell is Hamiltonian-connected if it conforms to a few practical restrictions.

A nondecreasing sequence of positive integers is $(\alpha,\beta)$-Conolly, or Conolly-like for short, if for every positive integer m the number of times that m occurs in the sequence is $\alpha + \beta r_m$, where $r_m$ is 1 plus the 2-adic valuation of m. A recurrence relation is $(\alpha, \beta)$-Conolly if it has an $(\alpha, \beta)$-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form $A(n) = \sum_{i=1}^k A(n-s_i-\sum_{j=1}^{p_i} A(n-a_{ij}))$ with appropriate initial conditions. For any fixed integers k and $p_1,p_2,\ldots, p_k$ we prove that there are only finitely many pairs $(\alpha, \beta)$ for which $A(n)$ can be $(\alpha, \beta)$-Conolly. For the case where $\alpha =0$ and $\beta =1$, we provide a bijective proof using labeled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence $H(n)=H(n-H(n-2)) + H(n-3-H(n-5))$ also has the Conolly sequence as a solution. When $k=2$ and $p_1=p_2$, we construct an example of an $(\alpha,\beta)$-Conolly recursion for every possible ($\alpha,\beta)$ pair, thereby providing the first examples of nested recursions with $p_i>1$ whose solutions are completely understood. Finally, in the case where $k=2$ and $p_1=p_2$, we provide an if and only if condition for a given nested recurrence $A(n)$ to be $(\alpha,0)$-Conolly by proving a very general ceiling function identity.