On the Area Requirements of Planar Greedy Drawings of Triconnected Planar Graphs

In this paper we study the area requirements of planar greedy drawings of triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family $\cal H$ of subdivisions of triconnected plane graphs and claimed that every planar greedy drawing of the graphs in $\mathcal H$ respecting the prescribed plane embedding requires exponential area. However, we show that every $n$-vertex graph in $\cal H$ actually has a planar greedy drawing respecting the prescribed plane embedding on an $O(n)\times O(n)$ grid. This reopens the question whether triconnected planar graphs admit planar greedy drawings on a polynomial-size grid. Further, we provide evidence for a positive answer to the above question by proving that every $n$-vertex Halin graph admits a planar greedy drawing on an $O(n)\times O(n)$ grid. Both such results are obtained by actually constructing drawings that are convex and angle-monotone. Finally, we consider $\alpha$-Schnyder drawings, which are angle-monotone and hence greedy if $\alpha\leq 30^\circ$, and show that there exist planar triangulations for which every $\alpha$-Schnyder drawing with a fixed $\alpha<60^\circ$ requires exponential area for any resolution rule

Overlay Addressing and Routing System Based on Hyperbolic Geometry

International audienceLocal knowledge routing schemes based on virtual coordinates taken from the hyperbolic plane have attracted considerable interest in recent years. In this paper, we propose a new approach for seizing the power of the hyperbolic geometry. We aim at building a scalable and reliable system for creating and managing overlay networks over the Internet. The system is implemented as a peer-to-peer infrastructure based on the transport layer connections between the peers. Through analysis, we show the limitations of the Poincaré disk model for providing virtual coordinates. Through simulations, we assess the practicability of our proposal. Results show that peer-to-peer overlays based on hyperbolic geometry have acceptable performances while introducing scalability and flexibility in dynamic peer-to-peer overlay networks

Greedy routing and virtual coordinates for future networks

At the core of the Internet, routers are continuously struggling with ever-growing routing and forwarding tables. Although hardware advances do accommodate such a growth, we anticipate new requirements e.g. in data-oriented networking where each content piece has to be referenced instead of hosts, such that current approaches relying on global information will not be viable anymore, no matter the hardware progress. In this thesis, we investigate greedy routing methods that can achieve similar routing performance as today but use much less resources and which rely on local information only. To this end, we add specially crafted name spaces to the network in which virtual coordinates represent the addressable entities. Our scheme enables participating routers to make forwarding decisions using only neighbourhood information, as the overarching pseudo-geometric name space structure already organizes and incorporates "vicinity" at a global level. A first challenge to the application of greedy routing on virtual coordinates to future networks is that of "routing dead-ends" that are local minima due to the difficulty of consistent coordinates attribution. In this context, we propose a routing recovery scheme based on a multi-resolution embedding of the network in low-dimensional Euclidean spaces. The recovery is performed by routing greedily on a blurrier view of the network. The different network detail-levels are obtained though the embedding of clustering-levels of the graph. When compared with higher-dimensional embeddings of a given network, our method shows a significant diminution of routing failures for similar header and control-state sizes. A second challenge to the application of virtual coordinates and greedy routing to future networks is the support of "customer-provider" as well as "peering" relationships between participants, resulting in a differentiated services environment. Although an application of greedy routing within such a setting would combine two very common fields of today's networking literature, such a scenario has, surprisingly, not been studied so far. In this context we propose two approaches to address this scenario. In a first approach we implement a path-vector protocol similar to that of BGP on top of a greedy embedding of the network. This allows each node to build a spatial map associated with each of its neighbours indicating the accessible regions. Routing is then performed through the use of a decision-tree classifier taking the destination coordinates as input. When applied on a real-world dataset (the CAIDA 2004 AS graph) we demonstrate an up to 40% compression ratio of the routing control information at the network's core as well as a computationally efficient decision process comparable to methods such as binary trees and tries. In a second approach, we take inspiration from consensus-finding in social sciences and transform the three-dimensional distance data structure (where the third dimension encodes the service differentiation) into a two-dimensional matrix on which classical embedding tools can be used. This transformation is achieved by agreeing on a set of constraints on the inter-node distances guaranteeing an administratively-correct greedy routing. The computed distances are also enhanced to encode multipath support. We demonstrate a good greedy routing performance as well as an above 90% satisfaction of multipath constraints when relying on the non-embedded obtained distances on synthetic datasets. As various embeddings of the consensus distances do not fully exploit their multipath potential, the use of compression techniques such as transform coding to approximate the obtained distance allows for better routing performances

Graph embeddings for low-stretch greedy routing

The simplest greedy geometric routing forwards packets to make most progress in terms of geometric distance within reach. Its notable advantages are low complexity, and the use of local information only. However, two problems of greedy routing are that delivery is not always guaranteed, and that the greedy routes may take more hops than the corresponding shortest paths. Additionally, in dynamic multihop networks, routing elements can join or leave during network operation or exhibit intermittent failures. Even a single link or node removal may invalidate the greedy routing success guarantees. Greedy embedding is a graph embedding that makes the simple greedy packet forwarding successful for every source-destination pair. In this dissertation, we consider the problems of designing greedy graph embeddings that also yield low hop stretch of the greedy paths over the shortest paths and can accommodate network dynamics. In the first part of the dissertation, we consider embedding and routing for arbitrary unweighted network graphs, based on greedy routing and utilizing virtual node coordinates. We propose an algorithm for online greedy graph embedding in the hyperbolic plane that enables incremental embedding of network nodes as they join the network, without disturbing the global embedding. As an alternative to frequent reembedding of temporally dynamic network graphs in order to retain the greedy embedding property, we propose a simple but robust generalization of greedy geometric routing called Gravity--Pressure (GP) routing. Our routing method always succeeds in finding a route to the destination provided that a path exists, even if a significant fraction of links or nodes is removed subsequent to the embedding. GP routing does not require precomputation or maintenance of special spanning subgraphs and is particularly suitable for operation in tandem with our proposed algorithm for online graph embedding. In the second part of the dissertation we study how topological and geometric properties of embedded graphs influence the hop stretch. Based on the obtained insights, we synthesize embedding heuristics that yield minimal hop stretch greedy embeddings. Finally, we verify their effectiveness on models of synthetic graphs as well as instances of several classes of real-world network graphs

Routing at Large Scale: Advances and Challenges for Complex Networks

International audienceA wide range of social, technological and communication systems can be described as complex networks. Scale-free networks are one of the well-known classes of complex networks in which nodes degree follow a power-law distribution. The design of scalable, adaptive and resilient routing schemes in such networks is very challenging. In this article we present an overview of required routing functionality, categorize the potential design dimensions of routing protocols among existing routing schemes and analyze experimental results and analytical studies performed so far to identify the main trends/trade-offs and draw main conclusions. Besides traditional schemes such as hierarchical/shortest-path path-vector routing, the article pays attention to advances in compact routing and geometric routing since they are known to significantly improve the scalability in terms of memory space. The identified trade-offs and the outcomes of this overview enable more careful conclusions regarding the (in-)suitability of different routing schemes to large-scale complex networks and provide a guideline for future routing research