The structure connectivity of Data Center Networks

Last decade, numerous giant data center networks are built to provide increasingly fashionable web applications. For two integers $m\geq 0$ and $n\geq 2$, the $m$-dimensional DCell network with $n$-port switches $D_{m,n}$ and $n$-dimensional BCDC network $B_{n}$ have been proposed. Connectivity is a basic parameter to measure fault-tolerance of networks. As generalizations of connectivity, structure (substructure) connectivity was recently proposed. Let $G$ and $H$ be two connected graphs. Let $\mathcal{F}$ be a set whose elements are subgraphs of $G$, and every member of $\mathcal{F}$ is isomorphic to $H$ (resp. a connected subgraph of $H$). Then $H$-structure connectivity $\kappa(G; H)$ (resp. $H$-substructure connectivity $\kappa^{s}(G; H)$) of $G$ is the size of a smallest set of $\mathcal{F}$ such that the rest of $G$ is disconnected or the singleton when removing $\mathcal{F}$. Then it is meaningful to calculate the structure connectivity of data center networks on some common structures, such as star $K_{1,t}$, path $P_k$, cycle $C_k$, complete graph $K_s$ and so on. In this paper, we obtain that $\kappa (D_{m,n}; K_{1,t})=\kappa^s (D_{m,n}; K_{1,t})=\lceil \frac{n-1}{1+t}\rceil+m$ for $1\leq t\leq m+n-2$ and $\kappa (D_{m,n}; K_s)= \lceil\frac{n-1}{s}\rceil+m$ for $3\leq s\leq n-1$ by analyzing the structural properties of $D_{m,n}$. We also compute $\kappa(B_n; H)$ and $\kappa^s(B_n; H)$ for $H\in \{K_{1,t}, P_{k}, C_{k}|1\leq t\leq 2n-3, 6\leq k\leq 2n-1 \}$ and $n\geq 5$ by using $g$-extra connectivity of $B_n$

Datacenter Traffic Control: Understanding Techniques and Trade-offs

Datacenters provide cost-effective and flexible access to scalable compute and storage resources necessary for today's cloud computing needs. A typical datacenter is made up of thousands of servers connected with a large network and usually managed by one operator. To provide quality access to the variety of applications and services hosted on datacenters and maximize performance, it deems necessary to use datacenter networks effectively and efficiently. Datacenter traffic is often a mix of several classes with different priorities and requirements. This includes user-generated interactive traffic, traffic with deadlines, and long-running traffic. To this end, custom transport protocols and traffic management techniques have been developed to improve datacenter network performance. In this tutorial paper, we review the general architecture of datacenter networks, various topologies proposed for them, their traffic properties, general traffic control challenges in datacenters and general traffic control objectives. The purpose of this paper is to bring out the important characteristics of traffic control in datacenters and not to survey all existing solutions (as it is virtually impossible due to massive body of existing research). We hope to provide readers with a wide range of options and factors while considering a variety of traffic control mechanisms. We discuss various characteristics of datacenter traffic control including management schemes, transmission control, traffic shaping, prioritization, load balancing, multipathing, and traffic scheduling. Next, we point to several open challenges as well as new and interesting networking paradigms. At the end of this paper, we briefly review inter-datacenter networks that connect geographically dispersed datacenters which have been receiving increasing attention recently and pose interesting and novel research problems.Comment: Accepted for Publication in IEEE Communications Surveys and Tutorial

Duality in algebra and topology

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the differential graded ring of cochains on a space, the differential graded ring of chains on the loop space, and various ring spectra, e.g., the Spanier-Whitehead duals of finite spectra or chromatic localizations of the sphere spectrum. Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities: Poincare duality for manifolds, Gorenstein duality for commutative rings, Benson-Carlson duality for cohomology rings of finite groups, Poincare duality for groups, Gross-Hopkins duality in chromatic stable homotopy theory, as examples of a single phenomenon. Beyond setting up this framework, though, we prove some new results, both in algebra and topology, and give new proofs of a number of old results