In this paper, we consider the partial gathering problem of mobile agents in synchronous dynamic bidirectional ring networks. When k agents are distributed in the network, the partial gathering problem requires, for a given positive integer g (< k), that agents terminate in a configuration such that either at least g agents or no agent exists at each node. So far, the partial gathering problem has been considered in static graphs. In this paper, we start considering partial gathering in dynamic graphs. As a first step, we consider this problem in 1-interval connected rings, that is, one of the links in a ring may be missing at each time step. In such networks, focusing on the relationship between the values of k and g, we fully characterize the solvability of the partial gathering problem and analyze the move complexity of the proposed algorithms when the problem can be solved. First, we show that the g-partial gathering problem is unsolvable when k <= 2g. Second, we show that the problem can be solved with O(n log g) time and the total number of O(gn log g) moves when 2g + 1 <= k <= 3g - 2. Third, we show that the problem can be solved with O(n) time and the total number of O(kn) moves when 3g - 1 <= k <= 8g - 4. Notice that since k = O(g) holds when 3g - 1 <= k <= 8g - 4, the move complexity O(kn) in this case can be represented also as O(gn). Finally, we show that the problem can be solved with O(n) time and the total number of O(gn) moves when k >= 8g - 3. These results mean that the partial gathering problem can be solved also in dynamic rings when k >= 2g + 1. In addition, agents require a total number of \Omega(gn) moves to solve the partial (resp., total) gathering problem. Thus, when k >= 3g - 1, agents can solve the partial gathering problem with the asymptotically optimal total number of O(gn) moves.

翻译：本文研究了同步动态双向环网络中移动代理的部分聚集问题。当k个代理分布于网络中时，部分聚集问题要求对于给定的正整数g (< k)，代理最终终止于满足每个节点上要么至少有g个代理、要么不存在代理的配置状态。迄今为止，部分聚集问题仅在静态图中被考虑。本文首次在动态图中研究部分聚集问题。作为第一步，我们考虑1-区间连通环（即环中每条时间步可能缺失一条链路）中的该问题。在此类网络中，聚焦于k与g的取值关系，我们完整刻画了部分聚集问题的可解性，并分析了问题可解时提出算法的移动复杂度。首先，我们证明当k ≤ 2g时，g-部分聚集问题不可解。其次，我们证明当2g+1 ≤ k ≤ 3g-2时，该问题可在O(n log g)时间内以总计O(gn log g)次移动求解。再次，我们证明当3g-1 ≤ k ≤ 8g-4时，该问题可在O(n)时间内以总计O(kn)次移动求解。注意到当3g-1 ≤ k ≤ 8g-4时，k = O(g)成立，因此此情况下的移动复杂度O(kn)亦可表示为O(gn)。最后，我们证明当k ≥ 8g-3时，该问题可在O(n)时间内以总计O(gn)次移动求解。这些结果表明当k ≥ 2g+1时，部分聚集问题在动态环中亦可求解。此外，代理解决部分（或完全）聚集问题至少需要总计Ω(gn)次移动。因此当k ≥ 3g-1时，代理能够以渐近最优的O(gn)次总移动数解决部分聚集问题。