3,449 research outputs found

    On Compact Routing for the Internet

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    While there exist compact routing schemes designed for grids, trees, and Internet-like topologies that offer routing tables of sizes that scale logarithmically with the network size, we demonstrate in this paper that in view of recent results in compact routing research, such logarithmic scaling on Internet-like topologies is fundamentally impossible in the presence of topology dynamics or topology-independent (flat) addressing. We use analytic arguments to show that the number of routing control messages per topology change cannot scale better than linearly on Internet-like topologies. We also employ simulations to confirm that logarithmic routing table size scaling gets broken by topology-independent addressing, a cornerstone of popular locator-identifier split proposals aiming at improving routing scaling in the presence of network topology dynamics or host mobility. These pessimistic findings lead us to the conclusion that a fundamental re-examination of assumptions behind routing models and abstractions is needed in order to find a routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802

    Compact Oblivious Routing

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    Oblivious routing is an attractive paradigm for large distributed systems in which centralized control and frequent reconfigurations are infeasible or undesired (e.g., costly). Over the last almost 20 years, much progress has been made in devising oblivious routing schemes that guarantee close to optimal load and also algorithms for constructing such schemes efficiently have been designed. However, a common drawback of existing oblivious routing schemes is that they are not compact: they require large routing tables (of polynomial size), which does not scale. This paper presents the first oblivious routing scheme which guarantees close to optimal load and is compact at the same time - requiring routing tables of polylogarithmic size. Our algorithm maintains the polylogarithmic competitive ratio of existing algorithms, and is hence particularly well-suited for emerging large-scale networks

    On Efficient Distributed Construction of Near Optimal Routing Schemes

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    Given a distributed network represented by a weighted undirected graph G=(V,E)G=(V,E) on nn vertices, and a parameter kk, we devise a distributed algorithm that computes a routing scheme in (n1/2+1/k+D)no(1)(n^{1/2+1/k}+D)\cdot n^{o(1)} rounds, where DD is the hop-diameter of the network. The running time matches the lower bound of Ω~(n1/2+D)\tilde{\Omega}(n^{1/2}+D) rounds (which holds for any scheme with polynomial stretch), up to lower order terms. The routing tables are of size O~(n1/k)\tilde{O}(n^{1/k}), the labels are of size O(klog2n)O(k\log^2n), and every packet is routed on a path suffering stretch at most 4k5+o(1)4k-5+o(1). Our construction nearly matches the state-of-the-art for routing schemes built in a centralized sequential manner. The previous best algorithms for building routing tables in a distributed small messages model were by \cite[STOC 2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but suffers from substantially larger routing tables of size O(n1/2+1/k)O(n^{1/2+1/k}), while the latter has sub-optimal running time of O~(min{(nD)1/2n1/k,n2/3+2/(3k)+D})\tilde{O}(\min\{(nD)^{1/2}\cdot n^{1/k},n^{2/3+2/(3k)}+D\})

    Scalable Routing Easy as PIE: a Practical Isometric Embedding Protocol (Technical Report)

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    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.Comment: This work has been previously published in IEEE ICNP'11. The present document contains an additional optional mechanism, presented in Section III-D, to further improve performance by using route asymmetry. It also contains new simulation result

    Distance labeling schemes for trees

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    We consider distance labeling schemes for trees: given a tree with nn nodes, label the nodes with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the distance in the tree between the two nodes. A lower bound by Gavoille et. al. (J. Alg. 2004) and an upper bound by Peleg (J. Graph Theory 2000) establish that labels must use Θ(log2n)\Theta(\log^2 n) bits\footnote{Throughout this paper we use log\log for log2\log_2.}. Gavoille et. al. (ESA 2001) show that for very small approximate stretch, labels use Θ(lognloglogn)\Theta(\log n \log \log n) bits. Several other papers investigate various variants such as, for example, small distances in trees (Alstrup et. al., SODA'03). We improve the known upper and lower bounds of exact distance labeling by showing that 14log2n\frac{1}{4} \log^2 n bits are needed and that 12log2n\frac{1}{2} \log^2 n bits are sufficient. We also give (1+ϵ1+\epsilon)-stretch labeling schemes using Θ(logn)\Theta(\log n) bits for constant ϵ>0\epsilon>0. (1+ϵ1+\epsilon)-stretch labeling schemes with polylogarithmic label size have previously been established for doubling dimension graphs by Talwar (STOC 2004). In addition, we present matching upper and lower bounds for distance labeling for caterpillars, showing that labels must have size 2lognΘ(loglogn)2\log n - \Theta(\log\log n). For simple paths with kk nodes and edge weights in [1,n][1,n], we show that labels must have size k1klogn+Θ(logk)\frac{k-1}{k}\log n+\Theta(\log k)

    Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method

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    We use the incompressibility method based on Kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies. In most models for routing, for almost all labeled graphs Θ(n2)\Theta (n^2) bits are necessary and sufficient for shortest path routing. By `almost all graphs' we mean the Kolmogorov random graphs which constitute a fraction of 11/nc1-1/n^c of all graphs on nn nodes, where c>0c > 0 is an arbitrary fixed constant. There is a model for which the average case lower bound rises to Ω(n2logn)\Omega(n^2 \log n) and another model where the average case upper bound drops to O(nlog2n)O(n \log^2 n). This clearly exposes the sensitivity of such bounds to the model under consideration. If paths have to be short, but need not be shortest (if the stretch factor may be larger than 1), then much less space is needed on average, even in the more demanding models. Full-information routing requires Θ(n3)\Theta (n^3) bits on average. For worst-case static networks we prove a Ω(n2logn)\Omega(n^2 \log n) lower bound for shortest path routing and all stretch factors <2<2 in some networks where free relabeling is not allowed.Comment: 19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appea
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