We present PIE, a scalable routing scheme that achieves 100 packet delivery and low path stretch. It is easy to implement in a distributed fashion and works well when costs are associated to links. Scalability is achieved by using virtual coordinates in a space of concise dimensionality, which enables greedy routing based only on local knowledge. PIE is a general routing scheme, meaning that it works on any graph. We focus however on the Internet, where routing scalability is an urgent concern. We show analytically and by using simulation that the scheme scales extremely well on Internet-like graphs. In addition, its geometric nature allows it to react efficiently to topological changes or failures by finding new paths in the network at no cost, yielding better delivery ratios than standard algorithms. The proposed routing scheme needs an amount of memory polylogarithmic in the size of the network and requires only local communication between the nodes. Although each node constructs its coordinates and routes packets locally, the path stretch remains extremely low, even lower than for centralized or less scalable state-of-the-art algorithms: PIE always finds short paths and often enough finds the shortest paths.
@cite_5 and @cite_24 find a bounded stretch of @math with @math coordinates for planar graphs @cite_5 and combinatorial unit disk graphs @cite_24 . For arbitrary graphs, the scheme of @cite_24 also provides a stretch of @math . However, these algorithms require a full, centralized knowledge of the topology in input.
@cite_24 @cite_5
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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. A new packet routing model proposed by the Internet Engineering Task Force is MultiProtocol Label Switching, or MPLS [B. Davie and Y. Rekhter, MPLS: Technology and Applications, Morgan Kaufmann (Elsevier), New York, 2000]. Instead of each router's parsing the packet network layer header and doing its lookups based on that analysis (as in much of conventional packet routing), MPLS ensures that the analysis of the header is performed just once. The packet is then assigned a stack of labels, where the labels are usually much smaller than the packet headers themselves. When a router receives a packet, it examines the label at the top of the label stack and makes the decision of where the packet is forwarded based solely on that label. It can pop the top label off the stack if it so desires, and can also push some new labels onto the stack, before forwarding the packet. This scheme has several advantages over conventional routing protocols, the two primary ones being (a) reduced amount of header analysis at intermediate routers, which allows for faster switching times, and (b) better traffic engineering capabilities and hence easier handling of quality of service issues. However, essentially nothing is known at a theoretical level about the performance one can achieve with this protocol, or about the intrinsic trade-offs in its use of resources. This paper initiates a theoretical study of MPLS protocols, and routing algorithms and lower bounds are given for a variety of situations. We first study the routing problem on the line, a case which is already nontrivial, and give routing protocols whose trade-offs are close to optimality. We then extend our results for paths to trees, and thence onto more general graphs. These routing algorithms on general graphs are obtained by finding a tree cover of a graph, i.e., a small family of subtrees of the graph such that, for each pair of vertices, one of the trees in the family contains an (almost-)shortest path between them. Our results show tree covers of logarithmic size for planar graphs and graphs with bounded separators, which may be of independent interest.