We study greedy routing over @math nodes placed in a ring, with the between two nodes defined to be the clockwise or the absolute distance between them along the ring. Such graphs arise in the context of modeling social networks and in routing networks for peer-to-peer systems. We construct the first network over @math nodes in which greedy routing takes @math hops in the worst-case, with @math out-going links per node. Our result has the first asymptotically optimal greedy routing complexity. Previous constructions required @math hops.
The construction demonstrates that we can indeed design networks in which greedy routing along these metrics has asymptotically optimal routing complexity. Our contribution is a family of networks that extends the Butterfly network family, so as to facilitate efficient greedy routing. With @math links per node, greedy routes are @math in the worst-case, which is asymptotically optimal. For @math , this beats the lower bound of @cite_18 on symmetric, randomized greedy routing networks (and it meets it for @math ). In the specific case of @math , our greedy routing achieves @math average route length.
@cite_18
2160405192 2158969254 2144253712 2118347867
Several peer-to-peer networks are based upon randomized graph topologies that permit efficient greedy routing, e. g., randomized hypercubes, randomized Chord, skip-graphs and constructions based upon small-world percolation networks. In each of these networks, a node has out-degree Θ(log n), where n denotes the total number of nodes, and greedy routing is known to take O(log n) hops on average. We establish lower-bounds for greedy routing for these networks, and analyze Neighbor-of-Neighbor (NoN)- greedy routing. The idea behind NoN, as the name suggests, is to take a neighbor's neighbors into account for making better routing decisions.The following picture emerges: Deterministic routing networks like hypercubes and Chord have diameter Θ(log n) and greedy routing is optimal. Randomized routing networks like randomized hypercubes, randomized Chord, and constructions based on small-world percolation networks, have diameter Θ(log n log log n) with high probability. The expected diameter of Skip graphs is also Θ(log n log log n). In all of these networks, greedy routing fails to find short routes, requiring Ω(log n) hops with high probability. Surprisingly, the NoN- greedy routing algorithm is able to diminish route-lengths to Θ(log n log log n) hops, which is asymptotically optimal. Greedy routing is a novel routing paradigm where messages are always forwarded to the neighbor that is closest to the destination. Our main result is a polynomial-time algorithm that embeds combinatorial unit disk graphs (CUDGs - a CUDG is a UDG without any geometric information) into O(log 2 n)- dimensional space, permitting greedy routing with constant stretch. To the best of our knowledge, this is the first greedy embedding with stretch guarantees for this class of networks. Our main technical contribution involves extracting, in polynomial time, a constant number of isometric and balanced tree separators from a given CUDG. We do this by extending the celebrated Lipton-Tarjan separator theorem for planar graphs to CUDGs. Our techniques extend to other classes of graphs; for example, for general graphs, we obtain an O(log n)-stretch greedy embedding into O(log 2 n)-dimensional space. The greedy embeddings constructed by our algorithm can also be viewed as a constant-stretch compact routing scheme in which each node is assigned an O(log 3 n)-bit label. To the best of our knowledge, this result yields the best known stretch-space trade-off for compact routing on CUDGs. Extensive simulations on random wireless networks indicate that the average routing overhead is about 10 ; only few routes have a stretch above 1.5. We investigate the construction of greedy embeddings in polylogarithmic dimensional Euclidian spaces in order to achieve scalable routing through geographic routing. We propose a practical algorithm which uses random projection to achieve greedy forwarding on a space of dimension O(log(n)) where nodes have coordinates of size O(log(n)), thus achieving greedy forwarding using a route table at each node of polylogarithmic size with respect to the number of nodes. We further improve this algorithm by using a quasi-greedy algorithm which ensures greedy forwarding works along a path-wise construction, allowing us to further reduce the dimension of the embedding. The proposed algorithm, denoted GLoVE-U, is fully distributed and practical to implement. We evaluate the performance using extensive simulations and show that our greedy forwarding algorithm delivers low path stretch and scales properly. We propose an embedding and routing scheme for arbitrary network connectivity graphs, based on greedy routing and utilizing virtual node coordinates. In dynamic multihop packet-switching communication networks, routing elements can join or leave during network operation or exhibit intermittent failures. We present 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. Even a single link or node removal may invalidate the greedy routing success guarantees in network embeddings based on an embedded spanning tree subgraph. 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 distance 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, as demonstrated by our numerical evaluation, is particularly suitable for operation in tandem with our proposed algorithm for online graph embedding.