A Near-Optimal Deterministic Distributed Synchronizer

We provide the first deterministic distributed synchronizer with near-optimal time complexity and message complexity overheads. Concretely, given any distributed algorithm $\mathcal{A}$ that has time complexity $T$ and message complexity $M$ in the synchronous message-passing model (subject to some care in defining the model), the synchronizer provides a distributed algorithm $\mathcal{A}'$ that runs in the asynchronous message-passing model with time complexity $T \cdot poly(\log n)$ and message complexity $(M+m)\cdot poly(\log n)$. Here, $n$ and $m$ denote the number of nodes and edges in the network, respectively. The synchronizer is deterministic in the sense that if algorithm $\mathcal{A}$ is deterministic, then so is algorithm $\mathcal{A}'$. Previously, only a randomized synchronizer with near-optimal overheads was known by seminal results of Awerbuch, Patt-Shamir, Peleg, and Saks [STOC 1992] and Awerbuch and Peleg [FOCS 1990]. We also point out and fix some inaccuracies in these prior works. As concrete applications of our synchronizer, we resolve some longstanding and fundamental open problems in distributed algorithms: we get the first asynchronous deterministic distributed algorithms with near-optimal time and message complexities for leader election, breadth-first search tree, and minimum spanning tree computations: these all have message complexity $\tilde{O}(m)$ message complexity. The former two have $\tilde{O}(D)$ time complexity, where $D$ denotes the network diameter, and the latter has $\tilde{O}(D+\sqrt{n})$ time complexity. All these bounds are optimal up to logarithmic factors. Previously all such near-optimal algorithms were either restricted to the synchronous setting or required randomization.