图上群体协议中的计数问题
Counting in Population Protocols on Graphs
AI总结：
针对图上匿名智能体组成的群体协议，提出无需图知识的精确计数协议，基于随机位采样与对数近似，还给出图大小属性计算的不可能性结果。
AI中文摘要：
我们研究群体协议中智能体数量的计数问题，其中智能体由底层图$G=(V,E)$连接，$|V|=n$个节点。每一步，随机调度器均匀选择一条边，关联节点进行状态转换。根据标准假设，智能体是相同且匿名的，即没有标识符。为打破对称性，每次交互中均匀随机指定一个智能体作为发起者。我们的大小计数协议使用$\tilde O(n)$个状态，以高概率在$O(B(G)\cdot\boldsymbol{\text{log}}^2(n) + L(G)\boldsymbol{\text{log}}(n))$次交互中稳定，其中$B(G)$是广播时间，$L(G)$是负载均衡时间。该协议基于新颖的协议，用于采样独立随机位（给定调度器确定发起者和响应者），以及以高概率将$\text{log}\thinspace n$近似到加性误差$O(\text{log}\thinspace\text{log}\thinspace n)$，后者使用$O(\text{poly}\thinspace\text{log}(n))$个状态和$O(B(G)\boldsymbol{\text{log}}^2 n)$次交互。这两个结果可能具有独立意义。精确计数的主协议需要唯一领导者，另外两个协议不需要，所有协议都不需要图$G$的任何知识。我们最后给出终止均匀群体协议的不可能性结果，这类协议用于计算图大小属性（如计数节点或确定奇偶性），包括有领导者和无领导者两种情况。
英文摘要：
We consider the problem of counting the number of agents in a population protocol where the agents are connected by an underlying graph $G=(V,E)$ with $|V|=n$ nodes. In each step, a random scheduler selects an edge uniformly at random, and the incident nodes make a state transition. As per standard assumptions, agents are identical and anonymous, that is, have no identifiers. To break symmetry, in each interaction one of the agents is declared as the initiator uniformly at random. Our size counting protocol uses $\tilde O(n)$ states and stabilizes in $O( B(G) \cdot \log^2(n) + L(G) \cdot \log(n))$ interactions with high probability, where $B(G)$ is the broadcast time and $L(G)$ is the load balancing time. Our protocol is based on novel protocols for sampling independent random bits (given that the scheduler determines an initiator and responder) and approximating $\log n$ up to an additive error of $O(\log \log n)$ with high probability. The latter uses $O(poly\log(n))$ states and $O(B(G)\cdot\log^2 n)$ interactions. Both results may be of independent interest. The main protocol for exact counting requires the presence of a unique leader, the other two do not. None of the protocols requires any knowledge about the graph $G$. We conclude with impossibility results for terminating uniform population protocols that compute graph-size properties (like counting nodes or determining parity) with and without a leader.
