13,428 research outputs found

    A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

    Full text link
    In 2001 Thorup and Zwick devised a distance oracle, which given an nn-vertex undirected graph and a parameter kk, has size O(kn1+1/k)O(k n^{1+1/k}). Upon a query (u,v)(u,v) their oracle constructs a (2k1)(2k-1)-approximate path Π\Pi between uu and vv. The query time of the Thorup-Zwick's oracle is O(k)O(k), and it was subsequently improved to O(1)O(1) by Chechik. A major drawback of the oracle of Thorup and Zwick is that its space is Ω(nlogn)\Omega(n \cdot \log n). Mendel and Naor devised an oracle with space O(n1+1/k)O(n^{1+1/k}) and stretch O(k)O(k), but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size O(n1+1/k)O(n^{1+1/k}), stretch O(k)O(k) and query time O(nϵ)O(n^\epsilon), for an arbitrarily small ϵ>0\epsilon > 0. In particular, our oracle can provide logarithmic stretch using linear size. Another variant of our oracle has size O(nloglogn)O(n \log\log n), polylogarithmic stretch, and query time O(loglogn)O(\log\log n). For unweighted graphs we devise a distance oracle with multiplicative stretch O(1)O(1), additive stretch O(β(k))O(\beta(k)), for a function β()\beta(\cdot), space O(n1+1/kβ)O(n^{1+1/k} \cdot \beta), and query time O(nϵ)O(n^\epsilon), for an arbitrarily small constant ϵ>0\epsilon >0. The tradeoff between multiplicative stretch and size in these oracles is far below girth conjecture threshold (which is stretch 2k12k-1 and size O(n1+1/k)O(n^{1+1/k})). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch β(k)\beta(k) and size O(n1+1/k)O(n^{1+1/k}). A similar type of tradeoff was exhibited by a construction of (1+ϵ,β)(1+\epsilon,\beta)-spanners due to Elkin and Peleg. However, so far (1+ϵ,β)(1+\epsilon,\beta)-spanners had no counterpart in the distance oracles' world. An important novel tool that we develop on the way to these results is a {distance-preserving path-reporting oracle}

    Compact routing on the Internet AS-graph

    Get PDF
    Compact routing algorithms have been presented as candidates for scalable routing in the future Internet, achieving near-shortest path routing with considerably less forwarding state than the Border Gateway Protocol. Prior analyses have shown strong performance on power-law random graphs, but to better understand the applicability of compact routing algorithms in the context of the Internet, they must be evaluated against real- world data. To this end, we present the first systematic analysis of the behaviour of the Thorup-Zwick (TZ) and Brady-Cowen (BC) compact routing algorithms on snapshots of the Internet Autonomous System graph spanning a 14 year period. Both algorithms are shown to offer consistently strong performance on the AS graph, producing small forwarding tables with low stretch for all snapshots tested. We find that the average stretch for the TZ algorithm increases slightly as the AS graph has grown, while previous results on synthetic data suggested the opposite would be true. We also present new results to show which features of the algorithms contribute to their strong performance on these graphs

    Improved Parallel Algorithms for Spanners and Hopsets

    Full text link
    We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving: * Linear work parallel algorithms that construct spanners with O(k)O(k) stretch and size O(n1+1/k)O(n^{1+1/k}) in unweighted graphs, and size O(n1+1/klogk)O(n^{1+1/k} \log k) in weighted graphs. * Hopsets that lead to the first parallel algorithm for approximating shortest paths in undirected graphs with O(m  polylog  n)O(m\;\mathrm{polylog}\;n) work

    Improved Purely Additive Fault-Tolerant Spanners

    Full text link
    Let GG be an unweighted nn-node undirected graph. A \emph{β\beta-additive spanner} of GG is a spanning subgraph HH of GG such that distances in HH are stretched at most by an additive term β\beta w.r.t. the corresponding distances in GG. A natural research goal related with spanners is that of designing \emph{sparse} spanners with \emph{low} stretch. In this paper, we focus on \emph{fault-tolerant} additive spanners, namely additive spanners which are able to preserve their additive stretch even when one edge fails. We are able to improve all known such spanners, in terms of either sparsity or stretch. In particular, we consider the sparsest known spanners with stretch 66, 2828, and 3838, and reduce the stretch to 44, 1010, and 1414, respectively (while keeping the same sparsity). Our results are based on two different constructions. On one hand, we show how to augment (by adding a \emph{small} number of edges) a fault-tolerant additive \emph{sourcewise spanner} (that approximately preserves distances only from a given set of source nodes) into one such spanner that preserves all pairwise distances. On the other hand, we show how to augment some known fault-tolerant additive spanners, based on clustering techniques. This way we decrease the additive stretch without any asymptotic increase in their size. We also obtain improved fault-tolerant additive spanners for the case of one vertex failure, and for the case of ff edge failures.Comment: 17 pages, 4 figures, ESA 201

    Compact Routing on Internet-Like Graphs

    Full text link
    The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing scheme delivering a nearly optimal local memory upper bound. Using both direct analysis and simulation, we calculate the stretch distribution of this routing scheme on random graphs with power-law node degree distributions, PkkγP_k \sim k^{-\gamma}. We find that the average stretch is very low and virtually independent of γ\gamma. In particular, for the Internet interdomain graph, γ2.1\gamma \sim 2.1, the average stretch is around 1.1, with up to 70% of paths being shortest. As the network grows, the average stretch slowly decreases. The routing table is very small, too. It is well below its upper bounds, and its size is around 50 records for 10410^4-node networks. Furthermore, we find that both the average shortest path length (i.e. distance) dˉ\bar{d} and width of the distance distribution σ\sigma observed in the real Internet inter-AS graph have values that are very close to the minimums of the average stretch in the dˉ\bar{d}- and σ\sigma-directions. This leads us to the discovery of a unique critical quasi-stationary point of the average TZ stretch as a function of dˉ\bar{d} and σ\sigma. The Internet distance distribution is located in a close neighborhood of this point. This observation suggests the analytical structure of the average stretch function may be an indirect indicator of some hidden optimization criteria influencing the Internet's interdomain topology evolution.Comment: 29 pages, 16 figure

    Pruning based Distance Sketches with Provable Guarantees on Random Graphs

    Full text link
    Measuring the distances between vertices on graphs is one of the most fundamental components in network analysis. Since finding shortest paths requires traversing the graph, it is challenging to obtain distance information on large graphs very quickly. In this work, we present a preprocessing algorithm that is able to create landmark based distance sketches efficiently, with strong theoretical guarantees. When evaluated on a diverse set of social and information networks, our algorithm significantly improves over existing approaches by reducing the number of landmarks stored, preprocessing time, or stretch of the estimated distances. On Erd\"{o}s-R\'{e}nyi graphs and random power law graphs with degree distribution exponent 2<β<32 < \beta < 3, our algorithm outputs an exact distance data structure with space between Θ(n5/4)\Theta(n^{5/4}) and Θ(n3/2)\Theta(n^{3/2}) depending on the value of β\beta, where nn is the number of vertices. We complement the algorithm with tight lower bounds for Erdos-Renyi graphs and the case when β\beta is close to two.Comment: Full version for the conference paper to appear in The Web Conference'1

    Towards Bypassing Lower Bounds for Graph Shortcuts

    Get PDF
    For a given (possibly directed) graph G, a hopset (a.k.a. shortcut set) is a (small) set of edges whose addition reduces the graph diameter while preserving desired properties from the given graph G, such as, reachability and shortest-path distances. The key objective is in optimizing the tradeoff between the achieved diameter and the size of the shortcut set (possibly also, the distance distortion). Despite the centrality of these objects and their thorough study over the years, there are still significant gaps between the known upper and lower bound results. A common property shared by almost all known shortcut lower bounds is that they hold for the seemingly simpler task of reducing the diameter of the given graph, D_G, by a constant additive term, in fact, even by just one! We denote such restricted structures by (D_G-1)-diameter hopsets. In this paper we show that this relaxation can be leveraged to narrow the current gaps, and in certain cases to also bypass the known lower bound results, when restricting to sparse graphs (with O(n) edges): - {Hopsets for Directed Weighted Sparse Graphs.} For every n-vertex directed and weighted sparse graph G with D_G ? n^{1/4}, one can compute an exact (D_G-1)-diameter hopset of linear size. Combining this with known lower bound results for dense graphs, we get a separation between dense and sparse graphs, hence shortcutting sparse graphs is provably easier. For reachability hopsets, we can provide (D_G-1)-diameter hopsets of linear size, for sparse DAGs, already for D_G ? n^{1/5}. This should be compared with the diameter bound of O?(n^{1/3}) [Kogan and Parter, SODA 2022], and the lower bound of D_G = n^{1/6} by [Huang and Pettie, {SIAM} J. Discret. Math. 2018]. - {Additive Hopsets for Undirected and Unweighted Graphs.} We show a construction of +24 additive (D_G-1)-diameter hopsets with linear number of edges for D_G ? n^{1/12} for sparse graphs. This bypasses the current lower bound of D_G = n^{1/6} obtained for exact (D_G-1)-diameter hopset by [HP\u2718]. For general graphs, the bound becomes D_G ? n^{1/6} which matches the lower bound of exact (D_G-1) hopsets implied by [HP\u2718]. We also provide new additive D-diameter hopsets with linear size, for any given diameter D. Altogether, we show that the current lower bounds can be bypassed by restricting to sparse graphs (with O(n) edges). Moreover, the gaps are narrowed significantly for any graph by allowing for a constant additive stretch
    corecore