A Distributed Algorithm for the Minimum Diameter Spanning Tree Problem

International audienceWe present a new algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph G=(V,E,\omega). As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algorithm to find an absolute center of G. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves O(|V|) time complexity and O(|V| |E|) message complexity

The Multicast Policy and Its Relationship to Replicated Data Placement

In this paper we consider the communication complexity of maintaining the replicas of a logical data-item, in a database distributed over a computer network. We propose a new method, called the minimum spanning tree write, by which a processor in the network should multicast a write of a logical data-item, to all the processors that store replicas of the item. Then we show that the minimum spanning tree write is optimal from the communication cost point of view. We also demonstrate that the method by which a write is multicast to all the replicas of a data-item, affects the optimal replication scheme of the item, i.e., at which processors in the network the replicas should be located. Therefore, next we consider the problem of determining an optimal replication scheme for a data item, assuming that each processor employs the minimum spanning tree write at run-time. The problem for general networks is shown NP-Complete, but we provide efficient algorithms to obtain an optimal allocation scheme for three common types of network topologies. They are completely-connected, tree, and ring networks. For these topologies, efficient algorithms are also provided for the case in which reliability considerations dictate a minimum number of replicas

Almost-Tight Distributed Minimum Cut Algorithms

We study the problem of computing the minimum cut in a weighted distributed message-passing networks (the CONGEST model). Let $\lambda$ be the minimum cut, $n$ be the number of nodes in the network, and $D$ be the network diameter. Our algorithm can compute $\lambda$ exactly in $O((\sqrt{n} \log^{*} n+D)\lambda^4 \log^2 n)$ time. To the best of our knowledge, this is the first paper that explicitly studies computing the exact minimum cut in the distributed setting. Previously, non-trivial sublinear time algorithms for this problem are known only for unweighted graphs when $\lambda\leq 3$ due to Pritchard and Thurimella's $O(D)$-time and $O(D+n^{1/2}\log^* n)$-time algorithms for computing $2$-edge-connected and $3$-edge-connected components. By using the edge sampling technique of Karger's, we can convert this algorithm into a $(1+\epsilon)$-approximation $O((\sqrt{n}\log^{*} n+D)\epsilon^{-5}\log^3 n)$-time algorithm for any $\epsilon>0$. This improves over the previous $(2+\epsilon)$-approximation $O((\sqrt{n}\log^{*} n+D)\epsilon^{-5}\log^2 n\log\log n)$-time algorithm and $O(\epsilon^{-1})$-approximation $O(D+n^{\frac{1}{2}+\epsilon} \mathrm{poly}\log n)$-time algorithm of Ghaffari and Kuhn. Due to the lower bound of $\Omega(D+n^{1/2}/\log n)$ by Das Sarma et al. which holds for any approximation algorithm, this running time is tight up to a $\mathrm{poly}\log n$ factor. To get the stated running time, we developed an approximation algorithm which combines the ideas of Thorup's algorithm and Matula's contraction algorithm. It saves an $\epsilon^{-9}\log^{7} n$ factor as compared to applying Thorup's tree packing theorem directly. Then, we combine Kutten and Peleg's tree partitioning algorithm and Karger's dynamic programming to achieve an efficient distributed algorithm that finds the minimum cut when we are given a spanning tree that crosses the minimum cut exactly once