1,580 research outputs found

    Distributed Approximation Algorithms for Weighted Shortest Paths

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    A distributed network is modeled by a graph having nn nodes (processors) and diameter DD. We study the time complexity of approximating {\em weighted} (undirected) shortest paths on distributed networks with a O(logn)O(\log n) {\em bandwidth restriction} on edges (the standard synchronous \congest model). The question whether approximation algorithms help speed up the shortest paths (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is well-understood while its weighted counterpart is fundamental problem in the area of distributed approximation algorithms and remains widely open. We present new algorithms for computing both single-source shortest paths (\sssp) and all-pairs shortest paths (\apsp) in the weighted case. Our main result is an algorithm for \sssp. Previous results are the classic O(n)O(n)-time Bellman-Ford algorithm and an O~(n1/2+1/2k+D)\tilde O(n^{1/2+1/2k}+D)-time (8klog(k+1)1)(8k\lceil \log (k+1) \rceil -1)-approximation algorithm, for any integer k1k\geq 1, which follows from the result of Lenzen and Patt-Shamir (STOC 2013). (Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use O~()\tilde O(\cdot) to hide the O(\poly\log n) term.) We present an O~(n1/2D1/4+D)\tilde O(n^{1/2}D^{1/4}+D)-time (1+o(1))(1+o(1))-approximation algorithm for \sssp. This algorithm is {\em sublinear-time} as long as DD is sublinear, thus yielding a sublinear-time algorithm with almost optimal solution. When DD is small, our running time matches the lower bound of Ω~(n1/2+D)\tilde \Omega(n^{1/2}+D) by Das Sarma et al. (SICOMP 2012), which holds even when D=Θ(logn)D=\Theta(\log n), up to a \poly\log n factor.Comment: Full version of STOC 201

    Distributed Exact Shortest Paths in Sublinear Time

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    The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n)O(n) time, where nn is the number of vertices in the input graph GG. Peleg and Rubinovich (FOCS'99) showed a lower bound of Ω~(D+n)\tilde{\Omega}(D + \sqrt{n}) for this problem, where DD is the hop-diameter of GG. Whether or not this problem can be solved in o(n)o(n) time when DD is relatively small is a major notorious open question. Despite intensive research \cite{LP13,N14,HKN15,EN16,BKKL16} that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires O((nlogn)5/6)O((n \log n)^{5/6}) time, for D=O(nlogn)D = O(\sqrt{n \log n}), and O(D1/3(nlogn)2/3)O(D^{1/3} \cdot (n \log n)^{2/3}) time, for larger DD. This running time is sublinear in nn in almost the entire range of parameters, specifically, for D=o(n/log2n)D = o(n/\log^2 n). For the all-pairs shortest paths problem, our algorithm requires O(n5/3log2/3n)O(n^{5/3} \log^{2/3} n) time, regardless of the value of DD. We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our algorithm computes a hopset G"G" of a skeleton graph GG' of GG without first computing GG' itself. We then conduct a Bellman-Ford exploration in GG"G' \cup G", while computing the required edges of GG' on the fly. As a result, our algorithm computes exactly those edges of GG' that it really needs, rather than computing approximately the entire GG'

    A Superstabilizing log(n)\log(n)-Approximation Algorithm for Dynamic Steiner Trees

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    In this paper we design and prove correct a fully dynamic distributed algorithm for maintaining an approximate Steiner tree that connects via a minimum-weight spanning tree a subset of nodes of a network (referred as Steiner members or Steiner group) . Steiner trees are good candidates to efficiently implement communication primitives such as publish/subscribe or multicast, essential building blocks for the new emergent networks (e.g. P2P, sensor or adhoc networks). The cost of the solution returned by our algorithm is at most logS\log |S| times the cost of an optimal solution, where SS is the group of members. Our algorithm improves over existing solutions in several ways. First, it tolerates the dynamism of both the group members and the network. Next, our algorithm is self-stabilizing, that is, it copes with nodes memory corruption. Last but not least, our algorithm is \emph{superstabilizing}. That is, while converging to a correct configuration (i.e., a Steiner tree) after a modification of the network, it keeps offering the Steiner tree service during the stabilization time to all members that have not been affected by this modification

    Latency Optimal Broadcasting in Noisy Wireless Mesh Networks

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    In this paper, we adopt a new noisy wireless network model introduced very recently by Censor-Hillel et al. in [ACM PODC 2017, CHHZ17]. More specifically, for a given noise parameter p[0,1],p\in [0,1], any sender has a probability of pp of transmitting noise or any receiver of a single transmission in its neighborhood has a probability pp of receiving noise. In this paper, we first propose a new asymptotically latency-optimal approximation algorithm (under faultless model) that can complete single-message broadcasting task in D+O(log2n)D+O(\log^2 n) time units/rounds in any WMN of size n,n, and diameter DD. We then show this diameter-linear broadcasting algorithm remains robust under the noisy wireless network model and also improves the currently best known result in CHHZ17 by a Θ(loglogn)\Theta(\log\log n) factor. In this paper, we also further extend our robust single-message broadcasting algorithm to kk multi-message broadcasting scenario and show it can broadcast kk messages in O(D+klogn+log2n)O(D+k\log n+\log^2 n) time rounds. This new robust multi-message broadcasting scheme is not only asymptotically optimal but also answers affirmatively the problem left open in CHHZ17 on the existence of an algorithm that is robust to sender and receiver faults and can broadcast kk messages in O(D+klogn+polylog(n))O(D+k\log n + polylog(n)) time rounds.Comment: arXiv admin note: text overlap with arXiv:1705.07369 by other author

    On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

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    In the problem of minimum connected dominating set with routing cost constraint, we are given a graph G=(V,E)G=(V,E), and the goal is to find the smallest connected dominating set DD of GG such that, for any two non-adjacent vertices uu and vv in GG, the number of internal nodes on the shortest path between uu and vv in the subgraph of GG induced by D{u,v}D \cup \{u,v\} is at most α\alpha times that in GG. For general graphs, the only known previous approximability result is an O(logn)O(\log n)-approximation algorithm (n=Vn=|V|) for α=1\alpha = 1 by Ding et al. For any constant α>1\alpha > 1, we give an O(n11α(logn)1α)O(n^{1-\frac{1}{\alpha}}(\log n)^{\frac{1}{\alpha}})-approximation algorithm. When α5\alpha \geq 5, we give an O(nlogn)O(\sqrt{n}\log n)-approximation algorithm. Finally, we prove that, when α=2\alpha =2, unless NPDTIME(npolylogn)NP \subseteq DTIME(n^{poly\log n}), for any constant ϵ>0\epsilon > 0, the problem admits no polynomial-time 2log1ϵn2^{\log^{1-\epsilon}n}-approximation algorithm, improving upon the Ω(logn)\Omega(\log n) bound by Du et al. (albeit under a stronger hardness assumption)

    The covert set-cover problem with application to Network Discovery

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    We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover of size OPTOPT within O(logN)O(\log N) factor with high probability using O(OPTlog2N)O(OPT \cdot \log^2 N) queries where NN is the input size. We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown nn-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem we present an O(log2n)O(\log^2 n)-competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known algorithm has a competitive ratio of Ω(nlogn)\Omega (\sqrt{n\log n}) and therefore our result achieves an exponential improvement

    Deterministic Communication in Radio Networks

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    In this paper we improve the deterministic complexity of two fundamental communication primitives in the classical model of ad-hoc radio networks with unknown topology: broadcasting and wake-up. We consider an unknown radio network, in which all nodes have no prior knowledge about network topology, and know only the size of the network nn, the maximum in-degree of any node Δ\Delta, and the eccentricity of the network DD. For such networks, we first give an algorithm for wake-up, based on the existence of small universal synchronizers. This algorithm runs in O(min{n,DΔ}lognlogΔloglogΔ)O(\frac{\min\{n, D \Delta\} \log n \log \Delta}{\log\log \Delta}) time, the fastest known in both directed and undirected networks, improving over the previous best O(nlog2n)O(n \log^2n)-time result across all ranges of parameters, but particularly when maximum in-degree is small. Next, we introduce a new combinatorial framework of block synchronizers and prove the existence of such objects of low size. Using this framework, we design a new deterministic algorithm for the fundamental problem of broadcasting, running in O(nlogDloglogDΔn)O(n \log D \log\log\frac{D \Delta}{n}) time. This is the fastest known algorithm for the problem in directed networks, improving upon the O(nlognloglogn)O(n \log n \log \log n)-time algorithm of De Marco (2010) and the O(nlog2D)O(n \log^2 D)-time algorithm due to Czumaj and Rytter (2003). It is also the first to come within a log-logarithmic factor of the Ω(nlogD)\Omega(n \log D) lower bound due to Clementi et al.\ (2003). Our results also have direct implications on the fastest \emph{deterministic leader election} and \emph{clock synchronization} algorithms in both directed and undirected radio networks, tasks which are commonly used as building blocks for more complex procedures
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