In the stochastic population protocol model, we are given a connected graph with $n$ nodes, and in every time step, a scheduler samples an edge of the graph uniformly at random and the nodes connected by this edge interact. A fundamental task in this model is stable leader election, in which all nodes start in an identical state and the aim is to reach a configuration in which (1) exactly one node is elected as leader and (2) this node remains as the unique leader no matter what sequence of interactions follows. On cliques, the complexity of this problem has recently been settled: time-optimal protocols stabilize in $\Theta(n \log n)$ expected steps using $\Theta(\log \log n)$ states, whereas protocols that use $O(1)$ states require $\Theta(n^2)$ expected steps. In this work, we investigate the complexity of stable leader election on general graphs. We provide the first non-trivial time lower bounds for leader election on general graphs, showing that, when moving beyond cliques, the complexity landscape of leader election becomes very diverse: the time required to elect a leader can range from $O(1)$ to $\Theta(n^3)$ expected steps. On the upper bound side, we first observe that there exists a protocol that is time-optimal on many graph families, but uses polynomially-many states. In contrast, we give a near-time-optimal protocol that uses only $O(\log^2n)$ states that is at most a factor $\log n$ slower. Finally, we show that the constant-state protocol of Beauquier et al. [OPODIS 2013] is at most a factor $n \log n$ slower than the fast polynomial-state protocol. Moreover, among constant-state protocols, this protocol has near-optimal average case complexity on dense random graphs.
翻译:在调查人口协议模式中,我们得到一个与美元节点相连的图表,并且在每一个时间步骤中,一个调度器都以随机和此边缘连接的节点互动来抽样显示图表的边缘。模型中的一个基本任务是稳定的领导人选举,所有节点都以相同的状态开始,目的是达到以下配置:(1) 完全选举出一个节点作为领导者, (2) 这个节点仍然是独特的领导者, 不论互动的顺序如何。 在 cliques上, 这个问题的复杂性最近已经得到解决: 时间- 最佳协议稳定在$- 美元(n\ log n) 和 美元(n) 的节点。 这个模型中, 使用美元(n) 的节点开始于一个相同的状态, 而使用美元(n2) 协议的节点是要达到一个配置。 在一般图表上, 我们提供了第一个非三次更低的节点, 显示, 当我们从冰点开始, 最复杂的领导人选举的节点会变得非常多样化。