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$D$isthehop-diameteroftheinput$n$-vertex$m$-edge graph, and with message complexity $O(m+n^{3\mathrm{/}2})$. 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}.