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}.