Section: New Results

Network Design and Management

Participants : Gianlorenzo D'Angelo, Jean-Claude Bermond, Khoa Phan, David Coudert, Frédéric Giroire, Joanna Moulierac, Nicolas Nisse, Ronan Pardo Soares, Stéphane Pérennes, Issam Tahiri.

Network Design

Network design is a very wide subject that concerns all kinds of networks. We mainly study telecommunications networks which can be either physical networks (backbone, access, wireless, ...) or virtual (logical) ones. The objective is to design a network able to route a (given, estimated, dynamic, ...) traffic under some constraints (e.g. capacity) and with some quality of service (QoS) requirements. Usually the traffic is expressed as a family of requests with parameters attached to them. In order to satisfy these requests, we need to find one (or many) path(s) between their end nodes. The set of paths is chosen according to the technology, the protocol or the QoS constraints. The design can be done at the conception of the network (i.e. when conceiving a virtual network in MPLS where we have to establish virtual paths) or to adapt the network to changes (failures, new link, updates of routers, variation of traffic, ...). Finally there are various optimization criteria which differ according to the point of view: for a network user they are related to his/her satisfaction (minimizing delays, increasing available bandwidth, ...), while for a network operator, economics criteria like minimizing deployment and operating costs are more important.

This very wide topic is addressed by a lot of academic and industrial teams in the world. Our approach is to attack these problems with tools from Discrete Mathematics.

All-Optical Label Switching, AOLS

All-Optical Label Switching (AOLS) is a promising technology that performs packet forwarding without any optical-electrical-optical conversions, thus speeding up the forwarding. However, the cost of this technology requires limiting the number of labels needed to ensure the forwarding when routing a set of requests using GMPLS technology. In particular, this prevents the usage of label swapping techniques.

We have studied the routing problem in this context using label stacking techniques. We have formalized the problem by associating to each routing strategy a logical hypergraph, called a hypergraph layout, whose hyperarcs are dipaths of the physical graph, called tunnels in GMPLS terminology. We defined a cost function for the hypergraph layout, depending on its total length plus its total hop count. Minimizing the cost of the design of an AOLS network can then be expressed as finding a minimum cost hypergraph layout. In [24] , we prove hardness results for the problem. On the other hand, we provide approximation algorithms, in particular an $O(logn)$-approximation for symmetric directed networks. We focused on the case where the physical network is a directed path, providing a polynomial-time dynamic programming algorithm first for one source, and then for a fixed number $k$ of sources running in time $O\left(n^{k+2}\right)$.

Protocols

IP multicast is a protocol that deals with group communications with the aim of reducing traffic redundancy in the network. However, due to difficulty in deployment and poor scalability with a large number of multicast groups, IP multicast is still not widely deployed nor used on the Internet. Recently, Xcast6 and Xcast6 Treemap, two network layer multicast protocols, have been proposed with complementary scaling properties to IP multicast: they support a very large number of active multicast sessions. However, the key limitation of these protocols is that they only support small multicast groups. To overcome this limitation, we have proposed the Xcast6 Treemap Island [59] , [60] , a hybrid model of Application Layer Multicast (ALM) and Xcast6 that can work for large multicast groups. We have shown the feasibility of our model by simulation and comparison with IP multicast and NICE protocols.

Congestion control is a distributed algorithm to share network bandwidth among competing users on the Internet. In the common case, quick response time for mice traffic (http traffic) is desired when mixed with elephant traffic (ftp traffic). The current approach using loss-based with Additive Increase, Multiplicative Decrease (AIMD) is too greedy and eventually, most of the network bandwidth would be consumed by elephant traffic. As a result, it causes longer response time for mice traffic because there is no room left at the routers. MaxNet is a new TCP congestion control architecture using an explicit signal to control transmission rate at the source node. In [60] , we show that MaxNet can control well the queue length at routers and therefore the response time to http traffic is several times faster than with TCP Reno/RED.

Shared Risk Link Group

The notion of Shared Risk Link Group, SRLG has been introduced to capture multiple correlated failures in a network. A SRLG is a set of links that fail simultaneously if a given event (risk) occurs. In such multiple failures scenario, the problem of Diverse Routing consists in finding two SRLG-disjoint paths between a pair of nodes. We consider in [42] , [66] such problem for localized failures, when all the links of a SRLG verify the star property i.e. when they are incident to the same node. We prove that in this case the problem is in general NP-complete and determine some polynomial cases.

Data Gathering in Radio Networks

We study the problem of gathering information from the nodes of a radio network into a central node. We model the network of possible transmissions by a graph and consider a binary model of interference in which two transmissions interfere if the distance in the graph from the sender of one transmission to the receiver of the other is $d_I$ or less. A round is a set of non-interfering transmissions. In [25] , we determine the exact number of rounds required to gather one piece of information from each node of a square two-dimensional grid into the central node. The even case uses a method based on linear programming duality to prove the lower bound, and sophisticated algorithms using the symmetry of the grid and non-shortest paths to establish the matching upper bound. We then generalize our results to hexagonal grids.

Other results on multi-interface networks were obtained outside of Mascotte  [30] , [31] , [55] .

Routing

The problem of finding and updating shortest paths in distributed networks is considered crucial in today's practical applications. In the recent past, there has been a renewed interest in designing new efficient distance-vector algorithms (e.g., the distributed Bellman-Ford method implemented in the routing information protocol, RIP) as an alternative to link-state solutions (e.g., open shortest path first, OSPF) for large-scale distributed networks such as the autonomous systems topology of the Internet.

This year, we have proposed a new loop-free distance-vector routing algorithm, called LFR (Loop Free Routing), which is able to update the shortest paths of a distributed network with $n$ nodes in fully dynamic scenarios [47] . We compared experimentally this new algorithm with DUAL, one of the most popular loop-free distance vector algorithms which is part of CISCO's EIGRP protocol. Our experiments on CAIDA IPv4 routed /24 topology dataset show that LFR out-performs DUAL in terms of memory requirements and number of messages.

We then proposed a new technique, called Distributed Computation Pruning (DCP) [48] , for reducing the total number of messages sent and the space occupancy per node of every distance-vector routing algorithm based on shortest paths. We have evaluated experimentally the combination of DCP with DUAL and with LFR. We have observed that these combinations lead to a significant gain both in terms of number of messages sent and memory requirements per node.

We have also considered routing problems arising in road networs. In particular, we have conducted a theoretical study of the graph-augmentation problem of adding shortcuts in order to speedup route planning techniques [23] . We studied the algorithmic complexity of the problem and proposed approximation algorithms for a special graph class. We have also investigated ILP-based exact approaches and show how to stochastically evaluate a given shortcut assignment on graphs that are too large to do so exactly.

Compact routing

With the constant increase of the number of routing entries in the Internet, the size of the routing tables stored at router nodes increases drastically. Routing schemes such as BGP are showing their limits in terms of update time, search time, cost of signaling, etc. and alternatives have to be proposed. In particular, compact routing schemes propose interesting trade-offs between the size of the routing tables and the quality of the routes. They also take advantage of the particular properties arising in large scale networks such as low (logarithmic) diameter and high clustering coefficient.

High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we proposed in [37] a routing scheme that computes short routes in the class of $k$-chordal graphs, i.e., graphs with no induced cycles of length more than $k$. Our routing scheme achieves an additive stretch of at most $k-1$, and the routing tables are computed with a distributed algorithm which uses messages of size $O(logn)$ and takes $O\left(D\right)$ time, where $D$ is the diameter of the network.

We also used cops-and-robber games (See Section  6.2.1.2 ) to propose the first compact routing scheme for $k$-chordal graphs using routing tables, addresses and headers of size $O(logn)$ bits and achieving an additive stretch of $O(klog\Delta )$ [58] , [57] , [77] . This scheme is based on a new structural decomposition for a graph class including $k$-chordal graphs: we proposed a quadratic algorithm that, given a graph $G$ and $k\ge 3$, either returns an induced cycle larger than $k$ in $G$, or computes a tree-decomposition of $G$, each bag of which contains a dominating path with at most $k-1$ vertices. We thus proved that any $k$-chordal graph with maximum degree $\Delta$ has treewidth at most $(k-1)(\Delta -1)+2$, improving the $O\left(\Delta {(\Delta -1)}^{k-3}\right)$ bound of Bodlaender and Thilikos (1997). Moreover, any graph admitting such a tree-decomposition has small hyperbolicity.

In addition, we have pursued our investigation of the kind of structural graph properties that can or cannot be deduced from local (partial) views of the network. Such knowledge is crucial for the design of routing schemes. To this end, we have exhibited a hierarchy of problems and distributed models of computation [40] .

Routing models evaluation

The evaluation of new routing models asks for large-scale and intensive simulations. However, existing routing models simulators such as DRMSim are limited in terms of the number of routing table entries it can dynamically process and control on a single computer. Therefore, we have conducted a feasibility study of the extension of DRMSim so as to support the Distributed Parallel Discrete Event paradigm [46] . We have studied several distribution models and their associated communication overhead. We have in particular evaluated the expected additional time (in hours) required by a distributed simulation of BGP (border gate protocol), the current interdomain routing protocol of the Internet, compared to its sequential simulation. We show that such a distributed simulation of BGP is possible with a reasonable time overhead.

Reconfiguration

In production networks, traffic evolution, failures and maintenance operations force to adapt regularly the current configuration of the network (virtual topology, routing of connections). The routing reconfiguration problem in WDM networks consists of scheduling the migration of established lightpaths from current routing to a new pre-computed one while minimizing service disruptions. We have shown in the past the relations between this problem and the graph searching problem and established NP-completeness and inapproximability results.

This year, we proved the monotonicity of the process strategy game [78] , the graph searching game modeling the routing reconfiguration problem. Then, we have investigated on the influence of physical layer impairment constraints on the reconfiguration problem [41] . Setting up a new wavelength in a fiber of a WDM network requires recalibrating the other wavelengths passing through this fiber. This induces a cost (e.g., time, energy, degradation of QoS) that depends nonlinearly on the number of wavelengths using the fiber. Therefore, the order in which requests are switched affects the total cost of the operation. We have studied the corresponding optimization problem by modeling the cost of switching a request as a non-linear function depending on the load of the links used by the new lightpath. We have proved that determining the optimal rerouting order is NP-complete for a 2-nodes network, established general lower and upper bounds, identified classes of instances where the problem can be solved in polynomial time, and proposed a heuristic algorithm.

Energy efficiency

Recently, energy-aware routing has gained increasing popularity in the networking research community. The idea is that traffic demands are aggregated over a subset of the network links, allowing other links to be turned off to save energy. We develop several methods to improve routing protocols for backbone, wireless and content delivery networks. Several studies exhibit that the traffic load of the routers only has a small influence on their energy consumption. Hence, the power consumption in networks is strongly related to the number of active network elements, such as interfaces, line cards, base chassis,... The goal thus is to find a routing that minimizes the (weighted) number of active network elements used when routing. In [62] , we exhibit that the power consumption can be reduced of approximately 33 MWh for a medium-sized backbone network.

In [54] , we propose GreenRE - a new energy-aware routing model with the support of the new technique of data redundancy elimination (RE). Based on real experiments on Orange Labs platform and on simulations on several network topologies, we show that GreenRE can gain further 30% energy savings in comparison with the traditional energy-aware routing model.

One of the new challenges facing research in wireless networks is the design of algorithms and protocols that are energy aware. In [33] , we use for the first time the evolving graph combinatorial model as a tool to prove an NP-Completeness result, namely that computing a Minimum Spanning Tree of a planar network in the presence of mobility is actually NP-Complete.

Recently, there is a trend to introduce content caches as an inherent capacity of network equipment, with the objective of improving the efficiency of content distribution and reducing network congestion. In [63] , we study the impact of using in-network caches and CDN cooperation on an energy-efficient routing: up to 23% of power can be saved in the backbone this way.

In [32] , we study the energy efficiency of the networking part of data centers, accounting for between 10-20% of the total power consumption. We proposed a novel approach, called VMPlanner, for power reduction in the virtualization-based data centers. The idea of VMPlanner is to optimize both virtual machine placement and traffic flow routing so as to turn off as many unneeded network elements as possible for power saving.

Finally, in [56] , [38] , we summarize the main research results of the last years for energy efficiency for backbone, wireless, cellular and content distribution networks and highlight the main challenges of the field. Results are given for two operator networks, considering power and traffic forecasts for 2020.