We study what deterministic distributed algorithms can compute on random input graphs in extremely weak models of distributed computing: all nodes are anonymous, and in each communication round, nodes broadcast a message to all their neighbors, receive a (multi)set of messages from their neighbors, and update their local state. These correspond to the SB and MB models introduced by Hella et al. [PODC 2012] and are strictly weaker than the standard port-numbering PN and LOCAL models. We investigate what can be computed almost surely on random input graphs. We give a one-round deterministic SB-algorithm using $O(\log n)$-bit messages that computes unique identifiers with high probability on anonymous networks sampled from $G(n,p)$, where $n^{\varepsilon-1} \le p \le 1/2$ and $\varepsilon>0$ is an arbitrarily small constant. This algorithm is inspired by canonical labeling techniques in graph isomorphism testing and can be used to "anonymize" existing distributed graph algorithms designed for the broadcast CONGEST and LOCAL models. In particular, we give a new anonymous algorithm that finds a triangle in $O(1/\varepsilon)$ rounds on the above input distribution. We also investigate computational power of natural analogs of "Monte Carlo" and "Las Vegas" distributed graph algorithms in the random graph setting, and establish some new collapse and hierarchy results. For example, our work shows the collapse of the weak model hierarchy of Hella et al. on $G(n,p)$, as apart from a vanishingly small fraction of input graphs, the SB model is as powerful as LOCAL.

翻译：我们研究了在极弱的分布式计算模型中，确定性分布式算法在随机输入图上能够计算什么：所有节点都是匿名的，在每个通信轮次中，节点向所有邻居广播一条消息，接收来自邻居的消息（多重集），并更新其本地状态。这些对应于Hella等人[PODC 2012]引入的SB和MB模型，并且严格弱于标准的端口编号PN和LOCAL模型。我们探讨了在随机输入图上几乎必然能计算的内容。我们提出了一种一轮确定性SB算法，使用$O(\log n)$比特消息，在从$G(n,p)$（其中$n^{\varepsilon-1} \le p \le 1/2$，$\varepsilon>0$为任意小常数）采样的匿名网络上，以高概率计算唯一标识符。该算法受图同构测试中规范标记技术的启发，可用于“匿名化”为广播CONGEST和LOCAL模型设计的现有分布式图算法。特别地，我们给出了一种新的匿名算法，能在上述输入分布下以$O(1/\varepsilon)$轮次找到三角形。我们还研究了随机图设置中“蒙特卡洛”和“拉斯维加斯”分布式图算法的自然对应版本的计算能力，并建立了一些新的坍塌和层次结果。例如，我们的工作表明Hella等人的弱模型层次在$G(n,p)$上坍塌，因为除了可忽略的极小部分输入图外，SB模型与LOCAL模型具有同等能力。