How to design graphs with low forwarding index and limited number of edges

International audienceThe (edge) forwarding index of a graph is the minimum, over all possible rout-ings of all the demands, of the maximum load of an edge. This metric is of a great interest since it captures the notion of global congestion in a precise way: the lesser the forwarding-index, the lesser the congestion. In this paper, we study the following design question: Given a number e of edges and a number n of vertices, what is the least congested graph that we can construct? and what forwarding-index can we achieve? Our problem has some distant similarities with the well-known (∆, D) problem, and we sometimes build upon results obtained on it. The goal of this paper is to study how to build graphs with low forwarding indices and to understand how the number of edges impacts the forwarding index. We answer here these questions for different families of graphs: general graphs, graphs with bounded degree, sparse graphs with a small number of edges by providing constructions, most of them asymptotically optimal. For instance, we provide an asymptotically optimal construction for (n, n + k) cubic graphs-its forwarding index is ∼ n 2 3k log 2 (k). Our results allow to understand how the forwarding-index drops when edges are added to a graph and also to determine what is the best (i.e least congested) structure with e edges. Doing so, we partially answer the practical problem that initially motivated our work: If an operator wants to power only e links of its network, in order to reduce the energy consumption (or wiring cost) of its networks, what should be those links and what performance can be expected

An Introduction to Temporal Graphs: An Algorithmic Perspective

A \emph{temporal graph} is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence $G_1,G_2\ldots,G_l$ of static graphs over the same (static) set of nodes $V$. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension in added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community

Collocation Games and Their Application to Distributed Resource Management

We introduce Collocation Games as the basis of a general framework for modeling, analyzing, and facilitating the interactions between the various stakeholders in distributed systems in general, and in cloud computing environments in particular. Cloud computing enables fixed-capacity (processing, communication, and storage) resources to be offered by infrastructure providers as commodities for sale at a fixed cost in an open marketplace to independent, rational parties (players) interested in setting up their own applications over the Internet. Virtualization technologies enable the partitioning of such fixed-capacity resources so as to allow each player to dynamically acquire appropriate fractions of the resources for unencumbered use. In such a paradigm, the resource management problem reduces to that of partitioning the entire set of applications (players) into subsets, each of which is assigned to fixed-capacity cloud resources. If the infrastructure and the various applications are under a single administrative domain, this partitioning reduces to an optimization problem whose objective is to minimize the overall deployment cost. In a marketplace, in which the infrastructure provider is interested in maximizing its own profit, and in which each player is interested in minimizing its own cost, it should be evident that a global optimization is precisely the wrong framework. Rather, in this paper we use a game-theoretic framework in which the assignment of players to fixed-capacity resources is the outcome of a strategic "Collocation Game". Although we show that determining the existence of an equilibrium for collocation games in general is NP-hard, we present a number of simplified, practically-motivated variants of the collocation game for which we establish convergence to a Nash Equilibrium, and for which we derive convergence and price of anarchy bounds. In addition to these analytical results, we present an experimental evaluation of implementations of some of these variants for cloud infrastructures consisting of a collection of multidimensional resources of homogeneous or heterogeneous capacities. Experimental results using trace-driven simulations and synthetically generated datasets corroborate our analytical results and also illustrate how collocation games offer a feasible distributed resource management alternative for autonomic/self-organizing systems, in which the adoption of a global optimization approach (centralized or distributed) would be neither practical nor justifiable.NSF (CCF-0820138, CSR-0720604, EFRI-0735974, CNS-0524477, CNS-052016, CCR-0635102); Universidad Pontificia Bolivariana; COLCIENCIAS–Instituto Colombiano para el Desarrollo de la Ciencia y la Tecnología "Francisco José de Caldas

Graph Problems arising from Wavelength-routing in All-optical Networks

International audienceThis paper surveys the theoretical results obtained for wavelength{routing all{optical networks, presents some new results and proposes several open problems. In all{optical networks the vast bandwidth available is utilized through wavelength division multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on di erent wavelengths. The information, once transmitted as light, reaches its destination without being converted to electronic form inbetween, thus reaching high communication speed. We consider both networks with arbitrary topologies and particular networks of practical interest

Compact routing for the future internet

The Internet relies on its inter-domain routing system to allow data transfer between any two endpoints regardless of where they are located. This routing system currently uses a shortest path routing algorithm (modified by local policy constraints) called the Border Gateway Protocol. The massive growth of the Internet has led to large routing tables that will continue to grow. This will present a serious engineering challenge for router designers in the long-term, rendering state (routing table) growth at this pace unsustainable. There are various short-term engineering solutions that may slow the growth of the inter-domain routing tables, at the expense of increasing the complexity of the network. In addition, some of these require manual configuration, or introduce additional points of failure within the network. These solutions may give an incremental, constant factor, improvement. However, we know from previous work that all shortest path routing algorithms require forwarding state that grows linearly with the size of the network in the worst case. Rather than attempt to sustain inter-domain routing through a shortest path routing algorithm, compact routing algorithms exist that guarantee worst-case sub-linear state requirements at all nodes by allowing an upper-bound on path length relative to the theoretical shortest path, known as path stretch. Previous work has shown the promise of these algorithms when applied to synthetic graphs with similar properties to the known Internet graph, but they haven't been studied in-depth on Internet topologies derived from real data. In this dissertation, I demonstrate the consistently strong performance of these compact routing algorithms for inter-domain routing by performing a longitudinal study of two compact routing algorithms on the Internet Autonomous System (AS) graph over time. I then show, using the k-cores graph decomposition algorithm, that the structurally important nodes in the AS graph are highly stable over time. This property makes these nodes suitable for use as the "landmark" nodes used by the most stable of the compact routing algorithms evaluated, and the use of these nodes shows similar strong routing performance. Finally, I present a decentralised compact routing algorithm for dynamic graphs, and present state requirements and message overheads on AS graphs using realistic simulation inputs. To allow the continued long-term growth of Internet routing state, an alternative routing architecture may be required. The use of the compact routing algorithms presented in this dissertation offer promise for a scalable future Internet routing system