We investigate space-time trade-offs for population protocols in sparse interaction graphs. In complete interaction graphs, optimal space-time trade-offs are known for the leader election and exact majority problems. However, it has remained open if other graph families exhibit similar space-time complexity trade-offs, as existing lower bound techniques do not extend beyond highly dense graphs. In this work, we show that -- unlike in complete graphs -- population protocols on bounded-degree trees do not exhibit significant asymptotic space-time trade-offs for leader election and exact majority. For these problems, we give constant-space protocols that have near-optimal worst-case expected stabilisation time. These new protocols achieve a linear speed-up compared to the state-of-the-art. Our results are based on two novel protocols, which we believe are of independent interest. First, we give a new fast self-stabilising 2-hop colouring protocol for general interaction graphs, whose stabilisation time we bound using a stochastic drift argument. Second, we give a self-stabilising tree orientation algorithm that builds a rooted tree in optimal time on any tree. As a consequence, we can use simple constant-state protocols designed for directed trees to solve leader election and exact majority fast. For example, we show that ``directed'' annihilation dynamics solve exact majority in $O(n^2 \log n)$ steps on directed trees.

翻译：本研究探讨稀疏交互图中群体协议的空间-时间权衡问题。在完全交互图中，领导者选举与精确多数问题的最优空间-时间权衡特性已得到充分研究。然而，由于现有下界技术无法扩展到高密度图之外，其他图族是否具有类似的空间-时间复杂度权衡特性仍悬而未决。本文证明——与完全图不同——有界度树上的群体协议在领导者选举和精确多数问题上并未展现出显著的渐近空间-时间权衡。针对这些问题，我们提出了具有近似最优最坏情况期望稳定时间的常数空间协议。这些新协议相比现有最优技术实现了线性加速。我们的结果基于两个新颖协议，其本身具有独立研究价值：首先，我们提出了一种适用于一般交互图的新型快速自稳定2跳着色协议，其稳定时间通过随机漂移论证进行界定；其次，我们提出了一种自稳定的树定向算法，可在任意树上以最优时间构建有根树。基于此，我们能够利用为有向树设计的简单常数状态协议快速解决领导者选举和精确多数问题。例如，我们证明"有向"湮灭动力学可在有向树上以$O(n^2 \log n)$步数解决精确多数问题。