Consensus is one of the most thoroughly studied problems in distributed computing, yet there are still complexity gaps that have not been bridged for decades. In particular, in the classical message-passing setting with processes' crashes, since the seminal works of Bar-Joseph and Ben-Or [1998] \cite{Bar-JosephB98} and Aspnes and Waarts [1996, 1998] \cite{AspnesW-SICOMP-96,Aspnes-JACM-98} in the previous century, there is still a fundamental unresolved question about communication complexity of fast randomized Consensus against a (strong) adaptive adversary crashing processes arbitrarily online. The best known upper bound on the number of communication bits is $Θ(\frac{n^{3/2}}{\sqrt{\log{n}}})$ per process, while the best lower bound is $Ω(1)$. This is in contrast to randomized Consensus against a (weak) oblivious adversary, for which time-almost-optimal algorithms guarantee amortized $O(1)$ communication bits per process \cite{GK-SODA-10}. We design an algorithm against adaptive adversary that reduces the communication gap by nearly linear factor to $O(\sqrt{n}\cdot\text{polylog } n)$ bits per process, while keeping almost-optimal (up to factor $O(\log^3 n)$) time complexity $O(\sqrt{n}\cdot\log^{5/2} n)$.
More surprisingly, we show this complexity indeed can be lowered further, but at the expense of increasing time complexity, i.e., there is a {\em trade-off} between communication complexity and time complexity. More specifically, our main Consensus algorithm allows to reduce communication complexity per process to any value from $\text{polylog } n$ to $O(\sqrt{n}\cdot\text{polylog } n)$, as long as Time $\times$ Communication $= O(n\cdot \text{polylog } n)$. Similarly, reducing time complexity requires more random bits per process, i.e., Time $\times$ Randomness $=O(n\cdot \text{polylog } n)$.
Efficient and Simple Algorithms for Fault Tolerant Spanners
最近表明，一个版本的贪心算法给出了一个结构的容错空间是大小最优的，至少对于顶点故障。然而，构造该时间序列的算法不是多项式时间的，并且最著名的多项式时间算法明显是次优的。设计一个多项式时间算法来构造（近）最优的容错空间是最近两篇关于容错空间的论文（[Bodwin，Dinitz，Parter，Vassilevka Williams SODA '18]和[Bodwin，Patel PODC '19]）中的一个公开问题。我们给出了一个令人惊讶的简单算法，在多项式时间内运行，并通过修改贪心算法运行在多项式时间，构造非常接近最优的容错spectrum（关闭只有一个线性因子的拉伸）。为了补充这一结果，我们还给出了简单的分布式结构的两个模型的CONGEST和CONGEST。
