Over the years, much research involving mobile computational entities has been performed. From modeling actual microscopic (and smaller) robots, to modeling software processes on a network, many important problems have been studied in this context. Gathering is one such fundamental problem in this area. The problem of gathering $k$ robots, initially arbitrarily placed on the nodes of an $n$-node graph, asks that these robots coordinate and communicate in a local manner, as opposed to global, to move around the graph, find each other, and settle down on a single node as fast as possible. A more difficult problem to solve is gathering with detection, where once the robots gather, they must subsequently realize that gathering has occurred and then terminate. In this paper, we propose a deterministic approach to solve gathering with detection for any arbitrary connected graph that is faster than existing deterministic solutions for even just gathering (without the requirement of detection) for arbitrary graphs. In contrast to earlier work on gathering, it leverages the fact that there are more robots present in the system to achieve gathering with detection faster than those previous papers that focused on just gathering. The state of the art solution for deterministic gathering~[Ta-Shma and Zwick, TALG, 2014] takes $\Tilde{O}$$(n^5 \log \ell)$ rounds, where $\ell$ is the smallest label among robots and $\Tilde{O}$ hides a polylog factor. We design a deterministic algorithm for gathering with detection with the following trade-offs depending on how many robots are present: (i) when $k \geq \lfloor n/2 \rfloor + 1$, the algorithm takes $O(n^3)$ rounds, (ii) when $k \geq \lfloor n/3 \rfloor + 1$, the algorithm takes $O(n^4 \log n)$ rounds, and (iii) otherwise, the algorithm takes $\Tilde{O}$$(n^5)$ rounds. The algorithm is not required to know $k$, but only $n$.

翻译：多年来，针对移动计算实体的研究已广泛展开。从模拟微观（乃至更小）的实际机器人，到模拟网络中的软件进程，这一背景下已研究了诸多重要问题。汇聚是该领域中一个基础性问题：初始随机分布于含n个节点的图上的k个机器人，需通过局部而非全局的协调与通信，在图中移动、相互寻找并尽快聚集到单一节点上。更具挑战性的问题是带检测的汇聚，即机器人完成聚集后还需意识到聚集已发生并终止运行。本文提出一种确定性方法，用以解决任意连通图上的带检测汇聚问题，其速度比现有任意图上的仅汇聚（无需检测）的确定性方案更快。与早期聚焦于仅汇聚的研究不同，本文利用系统中存在更多机器人的优势，实现了比先前工作更快的带检测汇聚。当前确定性汇聚的最优解决方案[Ta-Shma and Zwick, TALG, 2014]需$\Tilde{O}$$(n^5 \log \ell)$轮（其中$\ell$为机器人中的最小标签，$\Tilde{O}$隐藏多对数因子）。我们设计了一种带检测汇聚的确定性算法，根据机器人数量呈现如下权衡：（i）当$k \geq \lfloor n/2 \rfloor + 1$时，算法需$O(n^3)$轮；（ii）当$k \geq \lfloor n/3 \rfloor + 1$时，算法需$O(n^4 \log n)$轮；（iii）否则，算法需$\Tilde{O}$$(n^5)$轮。该算法无需知道k的具体值，仅需知道n。