近似对数时间内均匀布居协议中的精确大小计数
Exact size counting in uniform population protocols in nearly logarithmic time
摘要：
我们研究人口协议：网络的匿名代理下的调度程序，挑选对代理均匀随机互动。大小计数问题是计算人口中代理的确切数量$n$，假设没有领导者（每个代理都在相同的状态下开始）。我们给出了第一个协议，解决了这个问题在次线性时间。该协议在$O(\log n \log \log n)$时间内收敛，并以概率$1-O(\frac{\log \log n}{n})$使用$O(n^{60})$状态（每个代理的$O(1) + 60 \log n$内存位）。时间复杂度也是预期的$O(\log n \log \log n)$。收敛时间也是预期的$O(\log n \log \log n)$。重要的是，与大多数已发表的具有$ω(1)$状态的协议不同，我们的协议是_uniform_的：它对任何种群大小使用相同的转换算法，因此不需要将种群大小的估计嵌入到算法中。一个子协议是第一个统一的子线性时间的领导者选举人口协议，采取$O(\log n \log \log n)$时间和$O(n^{18})$状态。计数和领导选举协议的状态复杂度可以分别降低到$O(n^{30})$和$O(n^{9})$，同时增加时间到$O(\log^2 n)$。
英文摘要：
We study population protocols: networks of anonymous agents that interact under a scheduler that picks pairs of agents uniformly at random. The _size counting problem_ is that of calculating the exact number $n$ of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time. The protocol converges in $O(\log n \log \log n)$ time and uses $O(n^{60})$ states ($O(1) + 60 \log n$ bits of memory per agent) with probability $1-O(\frac{\log \log n}{n})$. The time complexity is also $O(\log n \log \log n)$ in expectation. The time to converge is also $O(\log n \log \log n)$ in expectation. Crucially, unlike most published protocols with $ω(1)$ states, our protocol is _uniform_: it uses the same transition algorithm for any population size, so does not need an estimate of the population size to be embedded into the algorithm. A sub-protocol is the first uniform sublinear-time leader election population protocol, taking $O(\log n \log \log n)$ time and $O(n^{18})$ states. The state complexity of both the counting and leader election protocols can be reduced to $O(n^{30})$ and $O(n^{9})$ respectively, while increasing the time to $O(\log^2 n)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
