We consider basic communication tasks in arbitrary radio networks: $k$-broadcasting and $k$-gathering. In the case of $k$-broadcasting messages from $k$ sources have to get to all nodes in the network. The goal of $k$-gathering is to collect messages from $k$ source nodes in a designated sink node. We consider these problems in the framework of distributed algorithms with advice. Krisko and Miller showed in 2021 that the optimal size of advice for $k$-broadcasting is $Θ(\min(\log Δ,$ $ \log k))$, where $Δ$ is equal to the maximum degree of a vertex of the input communication graph. We show that the same bound $Θ(\min(\log Δ, \log k))$ on the size of optimal labeling scheme holds also for the $k$-gathering problems. Moreover, we design fast algorithms for both problems with asymptotically optimal size of advice. For $k$-gathering our algorithm works in at most $D+k$ rounds, where $D$ is the diameter of the communication graph. This time bound is optimal even for centralized algorithms. We apply the $k$-gathering algorithm for $k$-broadcasting to achieve an algorithm working in time $O(D+\log^2 n+k)$ rounds. We also exhibit a logarithmic time complexity gap between distributed algorithms with advice of optimal size and distributed algorithms with distinct arbitrary labels.

翻译：我们研究任意无线电网络中的基本通信任务：k-广播与k-聚集。在k-广播场景中，来自k个源节点的消息需传递至网络中的所有节点；而k-聚集的目标是将k个源节点的消息收集至指定的汇聚节点。我们在带建议信息的分布式算法框架下探讨这些问题。Krisko与Miller于2021年证明，k-广播的最优建议信息规模为Θ(min(log Δ, log k))，其中Δ为输入通信图中顶点的最大度数。我们证明，对于k-聚集问题，最优标签方案的规模同样满足Θ(min(log Δ, log k))这一界限。此外，我们针对两个问题设计了具有渐近最优建议信息规模的快速算法。对于k-聚集，我们的算法至多在D+k轮内完成（D为通信图的直径），该时间界限即使对集中式算法也是最优的。我们将k-聚集算法应用于k-广播，实现了在O(D + log² n + k)轮内运行的算法。我们还揭示了具有最优规模建议信息的分布式算法与使用任意不同标签的分布式算法之间存在对数级时间复杂度差距。