We study leader election and population size counting for population protocols: networks of finite-state anonymous agents that interact randomly. We provide simple protocols for approximate counting of the size of the population and for leader election. We show a protocol for leader election that terminates in $O(\frac{{\log }^{2}n}{\log m})$ parallel time, where m is a parameter that belongs to $\mathrm{\Omega }(\log n)$ and $O(n)$, using $O(\max \{\log m,\log \log n\})$ bits. By adjusting m between $\log n$ and n, we obtain a leader election protocol whose time and space can be smoothly traded off between $O(\frac{{\log }^{2}n}{\log \log n})$ to $O(\log n)$ time and $O(\log \log n)$ to $O(\log n)$ bits. We also give a protocol which provides a constant factor approximation of $\log n$ of the population size n, or an upper bound $n_e$ which is at most $n^a$ for some constant $a>1$. This protocol assumes the existence of a unique leader and stabilizes in $\mathrm{\Theta }(\log n)$ parallel time.

我们研究领导者选举和人口协议的人口规模计数：随机交互的有限状态匿名代理网络。我们提供简单的协议来近似计算人口规模和领导选举。我们展示了一个领导人选举协议，该协议终止于$○(\frac{{\mathrm{日志}}^{2}n}{\mathrm{日志}米})$并行时间，其中m是属于的参数$\mathrm{\Omega }(\mathrm{日志}n)$和$○(n)$， 使用$○(\mathrm{最大限度}\{\mathrm{日志}米,\mathrm{日志}\mathrm{日志}n\})$位。通过调整m之间$\mathrm{日志}n$和n，我们得到一个领导人选举协议，其时间和空间可以在$○(\frac{{\mathrm{日志}}^{2}n}{\mathrm{日志}\mathrm{日志}n})$到$○(\mathrm{日志}n)$时间和$○(\mathrm{日志}\mathrm{日志}n)$到$○(\mathrm{日志}n)$位。我们还给出了一个协议，它提供了一个常数因子近似$\mathrm{日志}n$人口规模n或上限$n_e$最多是$n^{\mathrm{一种}}$对于一些常数$\mathrm{一种}>1$. 该协议假设存在唯一的领导者并稳定在$\mathrm{\theta }(\mathrm{日志}n)$并行时间。