Broadcast is a ubiquitous distributed computing problem that underpins many other system tasks. In static, connected networks, it was recently shown that broadcast is solvable without any node memory and only constant-size messages in worst-case asymptotically optimal time (Hussak and Trehan, PODC'19/STACS'20/DC'23). In the dynamic setting of adversarial topology changes, however, existing algorithms rely on identifiers, port labels, or polynomial memory to solve broadcast and compute functions over node inputs. We investigate space-efficient, terminating broadcast algorithms for anonymous, synchronous, 1-interval connected dynamic networks and introduce the first memory lower bounds in this setting. Specifically, we prove that broadcast with termination detection is impossible for idle-start algorithms (where only the broadcaster can initially send messages) and otherwise requires $\Omega(\log n)$ memory per node, where $n$ is the number of nodes in the network. Even if the termination condition is relaxed to stabilizing termination (eventually no additional messages are sent), we show that any idle-start algorithm must use $\omega(1)$ memory per node, separating the static and dynamic settings for anonymous broadcast. This lower bound is not far from optimal, as we present an algorithm that solves broadcast with stabilizing termination using $\mathcal{O}(\log n)$ memory per node in worst-case asymptotically optimal time. In sum, these results reveal the necessity of non-constant memory for nontrivial terminating computation in anonymous dynamic networks.

翻译：广播是一个普遍存在的分布式计算问题，支撑着许多其他系统任务。在静态连通网络中，最近的研究表明，广播可以在最坏情况下渐近最优时间内无需节点内存且仅使用恒定大小消息即可求解（Hussak与Trehan，PODC'19/STACS'20/DC'23）。然而，在对抗性拓扑变化的动态场景中，现有算法依赖于标识符、端口标签或多项式内存来解决广播问题并计算节点输入的函数。我们针对匿名、同步、1-区间连通的动态网络，研究空间高效的终止式广播算法，并首次在该场景下提出内存下界。具体而言，我们证明对于空闲启动算法（仅广播者初始可发送消息），带终止检测的广播是不可能的；否则每个节点需要$\Omega(\log n)$内存，其中$n$是网络中的节点数。即使将终止条件放宽至稳定化终止（最终不再发送额外消息），我们证明任何空闲启动算法必须使用$\omega(1)$内存，从而区分了匿名广播在静态与动态场景下的差异。该下界接近最优，因为我们提出了一种算法，能在最坏情况下渐近最优时间内使用每个节点$\mathcal{O}(\log n)$内存实现稳定化终止的广播。总之，这些结果揭示了在匿名动态网络中实现非平凡终止计算必须使用非常数内存。