Partial gathering of mobile agents in dynamic rings

TL;DR

The paper studies the $g$-partial gathering problem for $k$ mobile agents in synchronous 1-interval connected dynamic rings, where a final configuration must have either at least $g$ agents at each occupied node or none at a node. It provides a complete solvability characterization across four regimes of $k$ relative to $g$, along with phase-based algorithms that achieve tight time and move complexities: $O(n\log g)$ rounds and $O(gn\log g)$ moves for $2g+1\le k\le 3g-2$, $O(n)$ rounds and $O(kn)=O(gn)$ moves for $3g-1\le k\le 8g-4$, and $O(n)$ rounds and $O(gn)$ moves for $k\ge 8g-3$, with an initial unsolvability result for $k\le 2g$. The methods include selection, gathering, and, for the high-$k$ case, semi-selection/semi-gathering/achievement phases that exploit distinct IDs and chirality to bound moves. The results show that partial gathering is feasible in dynamic rings when $k\ge 2g+1$ and achieve asymptotic optimality in total moves for $k\ge 3g-1$, providing practical insights for scalable coordination under topology changes.

Abstract

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 Ω(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.

Figures (4)

• Figure 1: An example of the $g$-partial gathering problem in a dynamic ring ($g=3$).
• Figure 2: An execution example of the gathering phase when $2g+1\le k\le3g-2$ ($g=4$).
• Figure 3: An execution example of the gathering phase when $3g-1\le k\le 8g-4$ ($g=3$).
• Figure 4: An execution example of the achievement phase when $k\ge 8g-3$ ($g=3$).

Theorems & Definitions (24)

• Definition 1
• Theorem 1
• Theorem 2
• proof
• Theorem 3
• proof
• Lemma 1
• proof
• Lemma 2
• proof