在具有一位通信的匿名动态网络中进行计算
Computing in Anonymous Dynamic Networks with One-Bit Communications
AI总结：
研究匿名动态网络中一位通信的确定性计算，给出有领导者和无领导者时的算法及运行轮数，还得到下限，算法通过提取线性方程、割测试等从局部观测恢复多重性，保留了网络计算能力。
AI中文摘要：
我们开始研究匿名动态网络中的确定性计算，其中每个代理每轮广播一位，并且只接收广播每个位值的邻居数量。尽管有此严格限制，但仍可实现令人惊讶的丰富全局计算。在有唯一领导者且已知网络大小\(n\)的上限\(U\)时，对于大小为\(N = 2^{O(n\log n)}\)的全域中的输入多集的任何可计算函数，我们给出了一个在\(O(n^3\log^2 n + U)\)轮内终止的算法。在不知道\(n\)的情况下，我们为同一任务设计了一个在\(O(n^3\log^2 n)\)轮内运行的稳定算法。这基本上与拥塞模型的现有技术水平相匹配，在该模型中消息携带\(O(\log n)\)位且一般计算需要\(O(n^3)\)轮。我们还为无领导者和多领导者网络获得了可比结果。我们用一个几乎匹配的下限\(\Omega!\left(\frac{n^2\log(N/n)}{\log n}\right)\)轮补充了上限，当\(N = 2^{\Omega(n\log n)}\)时变为\(\Omega(n^3)\)。证明基于局部历史是信息论的，即使有唯一领导者、已知\(n\)和\(N\)以及限于动态变化环的通信图也成立。我们的算法从局部一位聚合观测中提取全局线性方程。一位割测试对不可区分代理类的大小产生守恒约束；通过细化这些类并收集独立约束，代理恢复所需的多重性。对于未知大小，我们引入了一个具有独立研究价值的自校正自适应泛洪原语。因此，即使每个消息都压缩为一位，拥塞匿名动态网络的计算能力也基本得以保留。
英文摘要：
We initiate the study of deterministic computation in anonymous dynamic networks where each agent broadcasts one bit per round and receives only the number of neighbors broadcasting each bit value. Despite this severe restriction, surprisingly rich global computation is possible. With a unique leader and a known upper bound $U$ on the network size $n$, we give a terminating algorithm for any computable function of the input multiset in $O(n^3\log^2 n+U)$ rounds, for inputs from a universe of size $N=2^{O(n\log n)}$. Without prior knowledge of $n$, we design a stabilizing algorithm for the same task running in $O(n^3\log^2 n)$ rounds. This essentially matches the state of the art for the congested model, where messages carry $O(\log n)$ bits and general computation takes $O(n^3)$ rounds. We also obtain comparable results for leaderless and multi-leader networks. We complement the upper bounds with an almost-matching lower bound of $$Ω\left(\frac{n^2\log(N/n)}{\log n}\right)$$ rounds, which becomes $Ω(n^3)$ for $N=2^{Ω(n\log n)}$. The proof is information-theoretic, based on local histories, and holds even with a unique leader, known $n$ and $N$, and a communication graph restricted to a dynamically changing ring. Our algorithms extract global linear equations from local one-bit aggregate observations. A one-bit cut test yields conservation constraints on the sizes of indistinguishable agent classes; by refining these classes and collecting independent constraints, agents recover the required multiplicities. For unknown size, we introduce a self-correcting adaptive flooding primitive of independent interest. Thus, the computational power of congested anonymous dynamic networks is essentially preserved even when every message is compressed to one bit.
