360 research outputs found

    Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating Set

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    This paper presents a near-optimal distributed approximation algorithm for the minimum-weight connected dominating set (MCDS) problem. The presented algorithm finds an O(logn)O(\log n) approximation in O~(D+n)\tilde{O}(D+\sqrt{n}) rounds, where DD is the network diameter and nn is the number of nodes. MCDS is a classical NP-hard problem and the achieved approximation factor O(logn)O(\log n) is known to be optimal up to a constant factor, unless P=NP. Furthermore, the O~(D+n)\tilde{O}(D+\sqrt{n}) round complexity is known to be optimal modulo logarithmic factors (for any approximation), following [Das Sarma et al.---STOC'11].Comment: An extended abstract version of this result appears in the proceedings of 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014

    Distributed Edge Connectivity in Sublinear Time

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    We present the first sublinear-time algorithm for a distributed message-passing network sto compute its edge connectivity λ\lambda exactly in the CONGEST model, as long as there are no parallel edges. Our algorithm takes O~(n11/353D1/353+n11/706)\tilde O(n^{1-1/353}D^{1/353}+n^{1-1/706}) time to compute λ\lambda and a cut of cardinality λ\lambda with high probability, where nn and DD are the number of nodes and the diameter of the network, respectively, and O~\tilde O hides polylogarithmic factors. This running time is sublinear in nn (i.e. O~(n1ϵ)\tilde O(n^{1-\epsilon})) whenever DD is. Previous sublinear-time distributed algorithms can solve this problem either (i) exactly only when λ=O(n1/8ϵ)\lambda=O(n^{1/8-\epsilon}) [Thurimella PODC'95; Pritchard, Thurimella, ACM Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14]. To achieve this we develop and combine several new techniques. First, we design the first distributed algorithm that can compute a kk-edge connectivity certificate for any k=O(n1ϵ)k=O(n^{1-\epsilon}) in time O~(nk+D)\tilde O(\sqrt{nk}+D). Second, we show that by combining the recent distributed expander decomposition technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup, STOC'15], we can decompose the network into a sublinear number of clusters with small average diameter and without any mincut separating a cluster (except the `trivial' ones). Finally, by extending the tree packing technique from [Karger STOC'96], we can find the minimum cut in time proportional to the number of components. As a byproduct of this technique, we obtain an O~(n)\tilde O(n)-time algorithm for computing exact minimum cut for weighted graphs.Comment: Accepted at 51st ACM Symposium on Theory of Computing (STOC 2019

    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

    Rational Fair Consensus in the GOSSIP Model

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    The \emph{rational fair consensus problem} can be informally defined as follows. Consider a network of nn (selfish) \emph{rational agents}, each of them initially supporting a \emph{color} chosen from a finite set Σ \Sigma. The goal is to design a protocol that leads the network to a stable monochromatic configuration (i.e. a consensus) such that the probability that the winning color is cc is equal to the fraction of the agents that initially support cc, for any cΣc \in \Sigma. Furthermore, this fairness property must be guaranteed (with high probability) even in presence of any fixed \emph{coalition} of rational agents that may deviate from the protocol in order to increase the winning probability of their supported colors. A protocol having this property, in presence of coalitions of size at most tt, is said to be a \emph{whp\,-tt-strong equilibrium}. We investigate, for the first time, the rational fair consensus problem in the GOSSIP communication model where, at every round, every agent can actively contact at most one neighbor via a \emph{push//pull} operation. We provide a randomized GOSSIP protocol that, starting from any initial color configuration of the complete graph, achieves rational fair consensus within O(logn)O(\log n) rounds using messages of O(log2n)O(\log^2n) size, w.h.p. More in details, we prove that our protocol is a whp\,-tt-strong equilibrium for any t=o(n/logn)t = o(n/\log n) and, moreover, it tolerates worst-case permanent faults provided that the number of non-faulty agents is Ω(n)\Omega(n). As far as we know, our protocol is the first solution which avoids any all-to-all communication, thus resulting in o(n2)o(n^2) message complexity.Comment: Accepted at IPDPS'1

    Beyond Geometry : Towards Fully Realistic Wireless Models

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    Signal-strength models of wireless communications capture the gradual fading of signals and the additivity of interference. As such, they are closer to reality than other models. However, nearly all theoretic work in the SINR model depends on the assumption of smooth geometric decay, one that is true in free space but is far off in actual environments. The challenge is to model realistic environments, including walls, obstacles, reflections and anisotropic antennas, without making the models algorithmically impractical or analytically intractable. We present a simple solution that allows the modeling of arbitrary static situations by moving from geometry to arbitrary decay spaces. The complexity of a setting is captured by a metricity parameter Z that indicates how far the decay space is from satisfying the triangular inequality. All results that hold in the SINR model in general metrics carry over to decay spaces, with the resulting time complexity and approximation depending on Z in the same way that the original results depends on the path loss term alpha. For distributed algorithms, that to date have appeared to necessarily depend on the planarity, we indicate how they can be adapted to arbitrary decay spaces. Finally, we explore the dependence on Z in the approximability of core problems. In particular, we observe that the capacity maximization problem has exponential upper and lower bounds in terms of Z in general decay spaces. In Euclidean metrics and related growth-bounded decay spaces, the performance depends on the exact metricity definition, with a polynomial upper bound in terms of Z, but an exponential lower bound in terms of a variant parameter phi. On the plane, the upper bound result actually yields the first approximation of a capacity-type SINR problem that is subexponential in alpha

    An Improved Distributed Algorithm for Maximal Independent Set

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    The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents an extremely simple randomized algorithm providing a near-optimal local complexity for this problem, which incidentally, when combined with some recent techniques, also leads to a near-optimal global complexity. Classical algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86] provide the global complexity guarantee that, with high probability, all nodes terminate after O(logn)O(\log n) rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node vv terminates after O(logdeg(v)+log1/ϵ)O(\log \mathsf{deg}(v)+\log 1/\epsilon) rounds, with probability at least 1ϵ1-\epsilon. The guarantee holds even if the randomness outside 22-hops neighborhood of vv is determined adversarially. This degree-dependency is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS'12, arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high probability global complexity of O(logΔ)+2O(loglogn)O(\log \Delta) + 2^{O(\sqrt{\log \log n})}, where Δ\Delta denotes the maximum degree. This improves over the O(log2Δ)+2O(loglogn)O(\log^2 \Delta) + 2^{O(\sqrt{\log \log n})} result of Barenboim et al., and gets close to the Ω(min{logΔ,logn})\Omega(\min\{\log \Delta, \sqrt{\log n}\}) lower bound of Kuhn et al. Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation Algorithms (LCA) model, and a faster distributed algorithm for the Lov\'asz Local Lemma
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