861 research outputs found

    Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees

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    In a directed graph GG with non-correlated edge lengths and costs, the \emph{network design problem with bounded distances} asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs. We give a bi-criteria (2+ε,O(n0.5+ε))(2+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for this problem. This improves on the currently best known linear approximation bound, at the cost of violating the distance bound by a factor of at most~2+ε2+\varepsilon. In the course of proving this result, the related problem of \emph{directed shallow-light Steiner trees} arises as a subproblem. In the context of directed graphs, approximations to this problem have been elusive. We present the first non-trivial result by proposing a (1+ε,O(Rε))(1+\varepsilon,O(|R|^{\varepsilon}))-ap\-proxi\-ma\-tion, where RR are the terminals. Finally, we show how to apply our results to obtain an (α+ε,O(n0.5+ε))(\alpha+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for \emph{light-weight directed α\alpha-spanners}. For this, no non-trivial approximation algorithm has been known before. All running times depends on nn and ε\varepsilon and are polynomial in nn for any fixed ε>0\varepsilon>0

    Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

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    Let GG be an nn-node and mm-edge positively real-weighted undirected graph. For any given integer f1f \ge 1, we study the problem of designing a sparse \emph{f-edge-fault-tolerant} (ff-EFT) σ\sigma{\em -approximate single-source shortest-path tree} (σ\sigma-ASPT), namely a subgraph of GG having as few edges as possible and which, following the failure of a set FF of at most ff edges in GG, contains paths from a fixed source that are stretched at most by a factor of σ\sigma. To this respect, we provide an algorithm that efficiently computes an ff-EFT (2F+1)(2|F|+1)-ASPT of size O(fn)O(f n). Our structure improves on a previous related construction designed for \emph{unweighted} graphs, having the same size but guaranteeing a larger stretch factor of 3(f+1)3(f+1), plus an additive term of (f+1)logn(f+1) \log n. Then, we show how to convert our structure into an efficient ff-EFT \emph{single-source distance oracle} (SSDO), that can be built in O~(fm)\widetilde{O}(f m) time, has size O(fnlog2n)O(fn \log^2 n), and is able to report, after the failure of the edge set FF, in O(F2log2n)O(|F|^2 \log^2 n) time a (2F+1)(2|F|+1)-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a \emph{minimum spanning forest} (MSF) of GG after that a \emph{batch} of kk simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in O(mlog3n)O(m \log^3 n) time a \emph{sensitivity} oracle of size O(mlog2n)O(m \log^2 n), that reports in O(k2log2n)O(k^2 \log^2 n) time the (at most 2k2k) edges either exiting from or entering into the MSF. [...]Comment: 16 pages, 4 figure

    Sparse Fault-Tolerant BFS Trees

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    This paper addresses the problem of designing a sparse {\em fault-tolerant} BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph TT of the given network GG such that subsequent to the failure of a single edge or vertex, the surviving part TT' of TT still contains a BFS spanning tree for (the surviving part of) GG. Our main results are as follows. We present an algorithm that for every nn-vertex graph GG and source node ss constructs a (single edge failure) FT-BFS tree rooted at ss with O(n \cdot \min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS tree rooted at ss. This result is complemented by a matching lower bound, showing that there exist nn-vertex graphs with a source node ss for which any edge (or vertex) FT-BFS tree rooted at ss has Ω(n3/2)\Omega(n^{3/2}) edges. We then consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees} for short, aiming to provide (following a failure) a BFS tree rooted at each source sSs\in S for some subset of sources SVS\subseteq V. Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every nn-vertex graph and source set SVS \subseteq V of size σ\sigma constructs a (single failure) FT-MBFS tree T(S)T^*(S) from each source siSs_i \in S, with O(σn3/2)O(\sqrt{\sigma} \cdot n^{3/2}) edges, and on the other hand there exist nn-vertex graphs with source sets SVS \subseteq V of cardinality σ\sigma, on which any FT-MBFS tree from SS has Ω(σn3/2)\Omega(\sqrt{\sigma}\cdot n^{3/2}) edges. Finally, we propose an O(logn)O(\log n) approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no Ω(logn)\Omega(\log n) approximation algorithm for these problems under standard complexity assumptions
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