Gathering Teams of Bounded Memory Agents on a Line

2025-07-05

AI Summary

This paper studies the gathering problem for deterministic finite-memory agents on an infinite undirected line graph: given a team of agents with unique identifiers, they must all converge to a single node and terminate. The problem is studied under a distributed synchronous model; analysis employs state-machine design and adversarial initial configuration to derive worst-case time complexity. The main contribution is a complete characterization of how team size fundamentally determines both feasibility and optimal gathering time. On the undirected line, for homogeneous teams (all agents share identical memory and capabilities), optimal time is Θ(D log L) for size-2 teams and improves to Θ(D) for size ≥ 3; heterogeneous teams achieve O(D) optimal gathering. These results establish team size as the critical threshold for breaking the log-factor bottleneck, revealing a size-driven complexity phase transition in distributed gathering.

Technology Category

• Multiagent Systems: Teamwork
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Several mobile agents, modelled as deterministic automata, navigate in an infinite line in synchronous rounds. All agents start in the same round. In each round, an agent can move to one of the two neighboring nodes, or stay idle. Agents have distinct labels which are integers from the set ${1,dots, L}$. They start in teams, and all agents in a team have the same starting node. The adversary decides the compositions of teams, and their starting nodes. Whenever an agent enters a node, it sees the entry port number and the states of all collocated agents; this information forms the input of the agent on the basis of which it transits to the next state and decides the current action. The aim is for all agents to gather at the same node and stop. Gathering is feasible, if this task can be accomplished for any decisions of the adversary, and its time is the worst-case number of rounds from the start till gathering. We consider the feasibility and time complexity of gathering teams of agents, and give a complete solution of this problem. It turns out that both feasibility and complexity of gathering depend on the sizes of teams. We first concentrate on the case when all teams have the same size $x$. For the oriented line, gathering is impossible if $x=1$, and it can be accomplished in time $O(D)$, for $x>1$, where $D$ is the distance between the starting nodes of the most distant teams. This complexity is of course optimal. For the unoriented line, the situation is different. For $x=1$, gathering is also impossible, but for $x=2$, the optimal time of gathering is $Θ(Dlog L)$, and for $xgeq 3$, the optimal time of gathering is $Θ(D)$. In the case when there are teams of different sizes, we show that gathering is always possible in time $O(D)$, even for the unoriented line. This complexity is of course optimal.

Problem

• Feasibility and time complexity of gathering bounded memory agents on a line
• Impact of team sizes on gathering feasibility and optimal time
• Optimal gathering strategies for oriented and unoriented lines with varying team sizes

Innovation

• Deterministic automata model synchronous agent movement
• Gathering feasibility depends on uniform team sizes
• Optimal time complexity achieved for diverse teams