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_4 adapts the scheme of for power-law graphs and obtains better scalability for the routing state, although still a fractional power of @math .
@cite_4
1527483699
We adapt the compact routing scheme by Thorup and Zwick to optimize it for power-law graphs. We analyze our adapted routing scheme based on the theory of unweighted random power-law graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bound coupled to the parameter of the power-law graph model for a compact routing scheme. In particular, we prove that, for stretch 3, instead of routing tables with O(n1 2) bits as in the general scheme by Thorup and Zwick, expected sizes of O(nγ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n1 + γ log n), with γ = τ - 2 2τ - 3 + Ɛ, where τ ∈ (2, 3) is the power-law exponent and Ɛ > 0. Both bounds also hold with probability at least 1 - 1 n (independent of Ɛ). The routing scheme is a labeled scheme, requiring a stretch-5 handshaking step and using addresses and message headers with O(log n log log n) bits, with probability at least 1 - o(1). We further demonstrate the effectiveness of our scheme by simulations on real-world graphs as well as synthetic power-law graphs. With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick for stretch 3 and obtain a new upper bound of expected O(n1+γ) for space and preprocessing.