Fast reroute paths algorithms

In order to keep services running despite link or node failure in MPLS networks, RSVP-TE fast reroute (FRR) schemes use precomputed backup label-switched path tunnels for local repair of LSP tunnels. In the event of failure, the redirection of traffic occurs onto backup LSP tunnels that have the same quality of service constraints as original paths. Local repair of LSP tunnels notably differ from traditional (1:1) dedicated path protection schemes in that traffic is diverted near the point of failure which speeds up the protection process by not having to notify the source and then resend the lost traffic. This gain in protection delay is crucial for MPLS networks which would otherwise suffer from an important recovery latency. In this paper, we investigate the algorithmic aspects of computing original paths along with their back-up so that they satisfy quality-of-service constraints (namely, delay) for single link or multiple link failure. In the case of single link failure, we propose an algorithm in O(nm+n 2log(n)) that computes shortest guaranteed paths with their backup towards a single destination. In the case of directed graphs, we show that this algorithm is optimal by proving that computing shortest guaranteed paths is as hard as to compute multiple source shortest paths in directed graphs. In the case of undirected graphs, we propose a faster algorithm with time complexity O(mlog(n)+n 2). We also provide a distributed algorithm based on Bellman-Ford distance computation which converges in 3n rounds at wors

The Four Principles of Geographic Routing

Geographic routing consists in using the position information of nodes to assist in the routing process, and has been a widely studied subject in sensor networks. One of the outstanding challenges facing geographic routing has been its applicability. Authors either make some broad assumptions on an idealized version of wireless networks which are often unverifiable, or they use costly methods to planarize the communication graph. The overarching questions that drive us are the following. When, and how should we use geographic routing? Is there a criterion to tell whether a communication network is fit for geographic routing? When exactly does geographic routing make sense? In this paper we formulate the four principles that define geographic routing and explore their topological consequences. Given a localized communication network, we then define and compute its geographic eccentricity, which measures its fitness for geographic routing. Finally we propose a distributed algorithm that either enables geographic routing on the network or proves that its geographic eccentricity is too high.Comment: This manuscript on geographic routing incoporates team feedback and expanded experiment

Virtual network embedding in the cycle

AbstractWe consider a problem motivated by the design of Asynchronous transfer mode (ATM) networks. Given a physical network and an all-to-all traffic, the problem consists in designing a virtual network with a given diameter, which can be embedded in the physical one with a minimum congestion (the congestion is the maximum load of a physical link). Here we solve the problem when the physical network is a ring. We give an almost optimal solution for diameter 2 and bounds for large diameters

Integral Symmetric 2-Commodity Flows

Let $G=(V,A)$ be a symmetric digraph, and $\kappa:A\rightarrow\mathbb{N}$ be a symmetric capacity. Let $s_1,s_2,t_1,t_2\in V$ and $v_1,v_2\in\mathbb{N}$. An integral symmetric 2-commodity flow in $G$ from $(s_1,s_2)$ to $(t_1,t_2)$ of value $(v_1,v_2)$ is an integral 4-commodity flow from $(s_1,t_1,s_2,t_2)$ to $(t_1,s_1,t_2,s_2)$ of value $(v_1,v_1,v_2,v_2)$. The Integral Symmetric 2-Commodity Flow Problem consists in finding a symmetric 2-commodity flow $(f_1,f_{-1},$ $f_2,f_{-2})$ from $(s_1,t_1)$ to $(s_2,t_2)$ of value $(v_1,v_2)$ such that $\Sigma f_i\leq\kappa$. It is known that the Integral 2-Commodity Flow Problem is NP-complete for both directed and undirected graphs (\cite{FHW80} and \cite{EIS76}). We prove that the cut criterion is a necessary and sufficient condition for the existence of a solution to the Integral Symmetric 2-Commodity Flow Problem, and give a polynomial-time algorithm in that provides a solution to this problem. The time complexity of our algorithm is \textbf{$6C_{flow}+O(|A|)$}, where $C_{flow}$ is the time complexity of your favorite flow algorithm (usually in $O(|V|\times|A|)$)

Two-Connected Graphs with Given Diameter

The problem we study in this work is an extremal problem arising from graph theory : what is the minimum number of edges of a 2-connected graph satisfying diameter conditions. This problem deals with survivable netowrk design when the network is subjected to satisfy grade of service constraints. One way to provide networks working when some failures arise is to provide a sufficient connectivity to the networks. Due to equipment robustness 2-connectivity or 2-edge-connectivity will be sufficient. Notice that k-connected networks, k$\geq$3, will provide too expensive networks. Another important parameter is the crossing-delay, ie the total amount of time spent in the network by some data packet to reach its destination from its origin. In order to keep this crossing-delay under reasonable values one can bound the number of hops of a routing path. This leads to bound the diameter of the underlying graph. We prove the following bounds : if $G$ is 2-(vertex)-connected, then |E|\geq\lceil\fracnD-(2D+1){D-1}\rceil, if $G$ is 2-edge-connected of odd diameter, then |E|\geq\lceil\fracnD-(2D+1){- D-1}\rceil, if $G$ is 2-edge-connected of even diameter, then |E|\geq min(\lceil\fracnD-(2D+1){D-1}\rceil,\lceil\frac{(n-1)(D+1)}{D}\rceil)

Computing shortest, fastest, and foremost journeys in dynamic networks

New technologies and the deployment of mobile and nomadic services are driving the emergence of complex communications networks, that have a highly dynamic behavior. This naturally engenders new route-discovery problems under changing conditions over these networks. Unfortunately, the temporal variations in the network topology are hard to be effectively captured in a classical graph model. In this paper, we use and extend a recently proposed graph theoretic model -- the evolving graphs --, which helps capture the evolving characteristic of such networks, in order to propose and formally analyze least cost journeys (the analog of paths in usual graphs) in a class of dynamic networks. Cost measures investigated here are hop count (shortest journeys), arrival date (foremost journeys), and time span (fastest journeys)