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

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 $O(D + \sqrt{n} \cdot \log^* n)$, where $D$ is the hop-diameter of the input $n$-vertex $m$-edge graph, and with message complexity $O(m + n^{3/2})$. 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 $\tilde{O}(D+ \sqrt{n})$ and message complexity $\tilde{O}(m)$. 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 $O((D + \sqrt{n}) \cdot \log n)$, using $O(m \cdot \log n + n \log n \cdot \log^* n)$ 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}

Tight Bounds For Distributed MST Verification

This paper establishes tight bounds for the Minimum-weight Spanning Tree (MST) verification problem in the distributed setting. Specifically, we provide an MST verification algorithm that achieves {\em simultaneously} $\tilde{O}(|E|)$ messages and $\tilde{O}(\sqrt{n} + D)$ time, where $|E|$ is the number of edges in the given graph $G$ and $D$ is $G$'s diameter. On the negative side, we show that any MST verification algorithm must send $\Omega(|E|)$ messages and incur $\tilde{\Omega}(\sqrt{n} + D)$ time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of $\Omega(|E|)$ messages and $\Omega(\sqrt{n} + D)$ time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves {\em simultaneously} $\tilde{O}(|E|)$ messages and $\tilde{O}(\sqrt{n} + D)$ time. Specifically, the best known time-optimal algorithm (using \tO(\sqrt{n} + D) time) requires $O(|E|+n^{3/2})$ messages, and the best known message-optimal algorithm (using \tO(|E|) messages) requires $O(n)$ time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction

Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

International audienceThis paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously O(m) messages and O(√ n+D) time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G's diameter. On the other hand, we show that any MST verification algorithm must send Ω(m) messages and incur Ω(√ n + D) time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of Ω(m) messages and Ω(√ n+D) time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously O(m) messages and O(√ n + D) time. Specifically, the best known time-optimal algorithm (using O(√ n + D) time) requires O(m + n 3/2) messages, and the best known message-optimal algorithm (using O(m) messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction

'Nachwort'

'Nachwort', in Helga Weissová, Zeichne, was Du siehst. Zeichnungen eines Kindes aus Theresienstadt, Frankfurt am Main and Leipzig: Insel Verlag, 2001, pp 94-111