在平均匿名网络中能计算什么？
What can be computed in average anonymous networks?
AI总结：
研究在随机输入图中，弱分布式计算模型下确定性分布式算法能解决的问题，提出一种使用O(log n)位消息的一轮SB算法，以高概率在匿名网络中计算唯一标识符，并发现弱模型在随机图中的崩溃。
摘要：
我们研究在随机输入图中，极弱的分布式计算模型下确定性分布式算法能计算什么。所有节点匿名，每个通信轮次中，节点向邻居广播消息，接收邻居的多集合消息并更新本地状态。这些对应于Hella等人[PODC 2012]引入的SB和MB模型，严格弱于标准端口编号PN和LOCAL模型。我们研究在随机输入图中几乎确定能计算什么。我们给出一种使用O(log n)位消息的一轮确定性SB算法，以高概率在从G(n,p)采样的匿名网络中计算唯一标识符，其中n^{ε-1} ≤ p ≤ 1/2且ε>0是任意小常数。该算法受图同构测试中的标准标记技术启发，可用于“匿名化”原有为广播CONGEST和LOCAL模型设计的分布式图算法。特别地，我们给出一种新的匿名算法，在上述输入分布上以O(1/ε)轮找到三角形。我们还研究了自然的“蒙特卡洛”和“拉斯维加斯”分布式图算法在随机图设置中的计算能力，并建立了某些新的崩溃和层级结果。例如，我们的工作显示Hella等人弱模型层级在G(n,p)上的崩溃，除了极小比例的输入图外，SB模型与LOCAL模型一样强大。
英文摘要：
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.
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