2,250 research outputs found

    Distributed Computing on Core-Periphery Networks: Axiom-based Design

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    Inspired by social networks and complex systems, we propose a core-periphery network architecture that supports fast computation for many distributed algorithms and is robust and efficient in number of links. Rather than providing a concrete network model, we take an axiom-based design approach. We provide three intuitive (and independent) algorithmic axioms and prove that any network that satisfies all axioms enjoys an efficient algorithm for a range of tasks (e.g., MST, sparse matrix multiplication, etc.). We also show the minimality of our axiom set: for networks that satisfy any subset of the axioms, the same efficiency cannot be guaranteed for any deterministic algorithm

    A Simple Deterministic Distributed MST Algorithm, with Near-Optimal Time and Message Complexities

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    Distributed minimum spanning tree (MST) problem is one of the most central and fundamental problems in distributed graph algorithms. Garay et al. \cite{GKP98,KP98} devised an algorithm with running time O(D+nβ‹…logβ‘βˆ—n)O(D + \sqrt{n} \cdot \log^* n), where DD is the hop-diameter of the input nn-vertex mm-edge graph, and with message complexity O(m+n3/2)O(m + n^{3/2}). Peleg and Rubinovich \cite{PR99} showed that the running time of the algorithm of \cite{KP98} is essentially tight, and asked if one can achieve near-optimal running time **together with near-optimal message complexity**. In a recent breakthrough, Pandurangan et al. \cite{PRS16} answered this question in the affirmative, and devised a **randomized** algorithm with time O~(D+n)\tilde{O}(D+ \sqrt{n}) and message complexity O~(m)\tilde{O}(m). They asked if such a simultaneous time- and message-optimality can be achieved by a **deterministic** algorithm. In this paper, building upon the work of \cite{PRS16}, we answer this question in the affirmative, and devise a **deterministic** algorithm that computes MST in time O((D+n)β‹…log⁑n)O((D + \sqrt{n}) \cdot \log n), using O(mβ‹…log⁑n+nlog⁑nβ‹…logβ‘βˆ—n)O(m \cdot \log n + n \log n \cdot \log^* n) messages. The polylogarithmic factors in the time and message complexities of our algorithm are significantly smaller than the respective factors in the result of \cite{PRS16}. Also, our algorithm and its analysis are very **simple** and self-contained, as opposed to rather complicated previous sublinear-time algorithms \cite{GKP98,KP98,E04b,PRS16}

    Distributed Connectivity Decomposition

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    We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity kk into (fractionally) vertex-disjoint weighted dominating trees with total weight Ξ©(klog⁑n)\Omega(\frac{k}{\log n}), in O~(D+n)\widetilde{O}(D+\sqrt{n}) rounds. (II) A decomposition of each undirected graph with edge-connectivity Ξ»\lambda into (fractionally) edge-disjoint weighted spanning trees with total weight βŒˆΞ»βˆ’12βŒ‰(1βˆ’Ξ΅)\lceil\frac{\lambda-1}{2}\rceil(1-\varepsilon), in O~(D+nΞ»)\widetilde{O}(D+\sqrt{n\lambda}) rounds. We also show round complexity lower bounds of Ξ©~(D+nk)\tilde{\Omega}(D+\sqrt{\frac{n}{k}}) and Ξ©~(D+nΞ»)\tilde{\Omega}(D+\sqrt{\frac{n}{\lambda}}) for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from O(n3)O(n^3) to near-optimal O~(m)\tilde{O}(m). As corollaries, we also get distributed oblivious routing broadcast with O(1)O(1)-competitive edge-congestion and O(log⁑n)O(\log n)-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal O(log⁑n)O(\log n)-approximation of vertex connectivity: centralized O~(m)\widetilde{O}(m) and distributed O~(D+n)\tilde{O}(D+\sqrt{n}). The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an O(m)O(m) centralized exact algorithm while the latter is the first distributed vertex connectivity approximation
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