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_10 constructs a fully distributed practical embedding by projecting an @math -dimensional graph topology onto a @math dimension Euclidean space using the Johnson-Lindenstrauss lemma. Despite attempting to preserve the relative distance between points, this method is -greedy and introduces some distortion in the embedded topology, which creates local minima. It therefore requires a recovery mechanism that significantly increases the path stretch.
@cite_10
2149463264 2037757210 1972939134 2144253712
The Johnson--Lindenstrauss lemma shows that any n points in Euclidean space (i.e., ℝn with distances measured under the e 2 norm) may be mapped down to O((log n) e2) dimensions such that no pairwise distance is distorted by more than a (1 p e) factor. Determining whether such dimension reduction is possible in e 1 has been an intriguing open question. We show strong lower bounds for general dimension reduction in e 1 . We give an explicit family of n points in e 1 such that any embedding with constant distortion D requires nΩ(1 D2) dimensions. This proves that there is no analog of the Johnson--Lindenstrauss lemma for e 1 ; in fact, embedding with any constant distortion requires nΩ(1) dimensions. Further, embedding the points into e 1 with (1pe) distortion requires nh−O(e log(1 e)) dimensions. Our proof establishes this lower bound for shortest path metrics of series-parallel graphs. We make extensive use of linear programming and duality in devising our bounds. We expect that the tools and techniques we develop will be useful for future investigations of embeddings into e 1 . A classic result of Johnson and Lindenstrauss asserts that any set of n points in d-dimensional Euclidean space can be embedded into k-dimensional Euclidean space---where k is logarithmic in n and independent of d--so that all pairwise distances are maintained within an arbitrarily small factor. All known constructions of such embeddings involve projecting the n points onto a spherically random k-dimensional hyperplane through the origin. We give two constructions of such embeddings with the property that all elements of the projection matrix belong in -1, 0, +1 . Such constructions are particularly well suited for database environments, as the computation of the embedding reduces to evaluating a single aggregate over k random partitions of the attributes. The Johnson-Lindenstrauss lemma provides a way to map a number of points in high-dimensional space into a low-dimensional space, with only a small distortion of the distances between the points. The proofs of the lemma are non-constructive: they show that a random mapping induces small distortions with high probability, but they do not construct the actual mapping. In this paper, we provide a procedure that constructs such a mapping deterministically in time almost linear in the number of distances to preserve times the dimension of the original space. We then use that result (together with Nisan's pseudorandom generator) to obtain an efficient derandomization of several approximation algorithms based on semidefinite programming. 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.