We study the classical Election problem in anonymous net- works, where solutions can rely on the use of random bits, which may be either shared or unshared among nodes. We provide a complete char- acterization of the conditions under which a randomized Election algo- rithm exists, for arbitrary structural knowledge. Our analysis considers both Las Vegas and Monte Carlo randomized algorithms, under the as- sumptions of shared and unshared randomness. In our setting, random sources are considered shared if the output bits are identical across spe- cific subsets of nodes. The algorithms and impossibility proofs are extensions of those of [5] for the deterministic setting. Our results are a complete generalization of those from [8]. Moreover, as applications, we consider many specific knowledge: no knowledge, a bound on the size, a bound on the number of nodes sharing a source, the size, or the full topology of the network. For each of them, we show how the general characterizations apply, showing they actually correspond to classes of structural knowledge. We also de- scribe also how randomized Election algorithms from the literature fits in this landscape. We therefore provide a comprehensive picture illustrating how knowledge influences the computability of the Election problem in arbitrary anonymous graphs with shared randomness.

翻译：本研究探讨匿名网络中的经典选举问题，其中解决方案可依赖于随机比特的使用，这些随机比特可能在节点间共享或非共享。针对任意结构知识，我们完整刻画了随机化选举算法存在的条件。我们的分析涵盖了拉斯维加斯和蒙特卡洛两类随机化算法，并分别在共享与非共享随机性的假设下展开讨论。在本研究设定中，若随机源输出比特在特定节点子集中保持一致，则被视为共享随机源。本文提出的算法与不可能性证明是对文献[5]中确定性场景研究结果的扩展。我们的结论完全推广了文献[8]的结果。此外，作为应用案例，我们考察了多种具体知识场景：无先验知识、网络规模上界、共享随机源的节点数量上界、确切网络规模以及完整网络拓扑。针对每种场景，我们展示了如何应用通用刻画方法，并证明它们实际上对应着不同的结构知识类别。我们还阐释了现有文献中的随机化选举算法如何融入这一理论框架。因此，本研究通过系统阐述知识如何影响具有共享随机性的任意匿名图中选举问题的可计算性，提供了该领域的完整理论图景。