42,358 research outputs found

    Fast and Compact Distributed Verification and Self-Stabilization of a DFS Tree

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    We present algorithms for distributed verification and silent-stabilization of a DFS(Depth First Search) spanning tree of a connected network. Computing and maintaining such a DFS tree is an important task, e.g., for constructing efficient routing schemes. Our algorithm improves upon previous work in various ways. Comparable previous work has space and time complexities of O(nlogΔ)O(n\log \Delta) bits per node and O(nD)O(nD) respectively, where Δ\Delta is the highest degree of a node, nn is the number of nodes and DD is the diameter of the network. In contrast, our algorithm has a space complexity of O(logn)O(\log n) bits per node, which is optimal for silent-stabilizing spanning trees and runs in O(n)O(n) time. In addition, our solution is modular since it utilizes the distributed verification algorithm as an independent subtask of the overall solution. It is possible to use the verification algorithm as a stand alone task or as a subtask in another algorithm. To demonstrate the simplicity of constructing efficient DFS algorithms using the modular approach, We also present a (non-sielnt) self-stabilizing DFS token circulation algorithm for general networks based on our silent-stabilizing DFS tree. The complexities of this token circulation algorithm are comparable to the known ones

    Improved Parallel Algorithms for Spanners and Hopsets

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    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

    Polynomial-Time Space-Optimal Silent Self-Stabilizing Minimum-Degree Spanning Tree Construction

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    Motivated by applications to sensor networks, as well as to many other areas, this paper studies the construction of minimum-degree spanning trees. We consider the classical node-register state model, with a weakly fair scheduler, and we present a space-optimal \emph{silent} self-stabilizing construction of minimum-degree spanning trees in this model. Computing a spanning tree with minimum degree is NP-hard. Therefore, we actually focus on constructing a spanning tree whose degree is within one from the optimal. Our algorithm uses registers on O(logn)O(\log n) bits, converges in a polynomial number of rounds, and performs polynomial-time computation at each node. Specifically, the algorithm constructs and stabilizes on a special class of spanning trees, with degree at most OPT+1OPT+1. Indeed, we prove that, unless NP == coNP, there are no proof-labeling schemes involving polynomial-time computation at each node for the whole family of spanning trees with degree at most OPT+1OPT+1. Up to our knowledge, this is the first example of the design of a compact silent self-stabilizing algorithm constructing, and stabilizing on a subset of optimal solutions to a natural problem for which there are no time-efficient proof-labeling schemes. On our way to design our algorithm, we establish a set of independent results that may have interest on their own. In particular, we describe a new space-optimal silent self-stabilizing spanning tree construction, stabilizing on \emph{any} spanning tree, in O(n)O(n) rounds, and using just \emph{one} additional bit compared to the size of the labels used to certify trees. We also design a silent loop-free self-stabilizing algorithm for transforming a tree into another tree. Last but not least, we provide a silent self-stabilizing algorithm for computing and certifying the labels of a NCA-labeling scheme

    A Faster Distributed Single-Source Shortest Paths Algorithm

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    We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of Ω~(n+D) \tilde \Omega (\sqrt{n} + D) rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where n n is the number of nodes in the network and D D is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs O~(n2/3D1/3+n5/6) \tilde O(n^{2/3} D^{1/3} + n^{5/6}) rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing O~(nD) \tilde O (\sqrt{n D}) rounds and the second performing O~(nD1/4+n3/5+D) \tilde O (\sqrt{n} D^{1/4} + n^{3/5} + D) rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a (1+ϵ) (1 + \epsilon) -approximation O~((nD1/4+D)/ϵ) \tilde O ((\sqrt{n} D^{1/4} + D) / \epsilon) -round algorithm for directed SSSP and a new work/depth trade-off for exact SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018

    Storage and Search in Dynamic Peer-to-Peer Networks

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    We study robust and efficient distributed algorithms for searching, storing, and maintaining data in dynamic Peer-to-Peer (P2P) networks. P2P networks are highly dynamic networks that experience heavy node churn (i.e., nodes join and leave the network continuously over time). Our goal is to guarantee, despite high node churn rate, that a large number of nodes in the network can store, retrieve, and maintain a large number of data items. Our main contributions are fast randomized distributed algorithms that guarantee the above with high probability (whp) even under high adversarial churn: 1. A randomized distributed search algorithm that (whp) guarantees that searches from as many as no(n)n - o(n) nodes (nn is the stable network size) succeed in O(logn){O}(\log n)-rounds despite O(n/log1+δn){O}(n/\log^{1+\delta} n) churn, for any small constant δ>0\delta > 0, per round. We assume that the churn is controlled by an oblivious adversary (that has complete knowledge and control of what nodes join and leave and at what time, but is oblivious to the random choices made by the algorithm). 2. A storage and maintenance algorithm that guarantees (whp) data items can be efficiently stored (with only Θ(logn)\Theta(\log{n}) copies of each data item) and maintained in a dynamic P2P network with churn rate up to O(n/log1+δn){O}(n/\log^{1+\delta} n) per round. Our search algorithm together with our storage and maintenance algorithm guarantees that as many as no(n)n - o(n) nodes can efficiently store, maintain, and search even under O(n/log1+δn){O}(n/\log^{1+\delta} n) churn per round. Our algorithms require only polylogarithmic in nn bits to be processed and sent (per round) by each node. To the best of our knowledge, our algorithms are the first-known, fully-distributed storage and search algorithms that provably work under highly dynamic settings (i.e., high churn rates per step).Comment: to appear at SPAA 201
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