When Robots Crash: Optimal Asynchronous Gathering at Weber Meeting Nodes

2026-10-03

AI Summary

This study addresses the optimal gathering problem for asynchronous, anonymous, and oblivious mobile robots on infinite grids subject to up to n−2 indistinguishable crash faults. It proposes the CrashTolerantWeberGathering algorithm, which departs from the conventional paradigm of relying on designated robots for symmetry breaking. Instead, each robot independently and consistently selects a Weber node as its target from a global snapshot and moves along constrained shortest paths. Fault tolerance is achieved by integrating strong multiplicity detection with Manhattan distance optimization. The authors rigorously prove the impossibility of optimal gathering under specific fully symmetric configurations while providing a complete solution that guarantees convergence to a Weber node in finite time for all remaining configurations, thereby significantly enhancing the fault-tolerant robustness of the system.

Abstract

We study the \textit{optimal gathering} problem over a finite set of designated \textit{meeting nodes} for \textit{asynchronous, anonymous,} and \textit{oblivious} mobile robots on an infinite grid under crash faults. The robots have global visibility and strong multiplicity detection, but share neither a coordinate system nor chirality. The objective is to gather all non-faulty robots at a \textsc{Weber Meeting Node}, minimizing the total Manhattan distance from their initial positions. Up to $n-2$ robots may crash permanently, and such crashes are indistinguishable from arbitrary delays. Existing approaches often rely on a designated robot to break symmetry, whose crash may block the remaining robots indefinitely. Instead, our approach enables every robot to independently select the same target from its snapshot, while target-dependent restricted shortest paths preserve the target as a \textsc{Weber Meeting Node}. We prove that, under strong multiplicity detection, optimal gathering is impossible from certain fully symmetric configurations. For all remaining configurations, our algorithm \textsc{CrashTolerantWeberGathering()} selects a unique common target, preserves its optimality throughout the execution, and allows non-faulty robots to progress without waiting for crashed robots, thereby guaranteeing gathering in finite time.

Problem

• Optimal Gathering
• Crash Faults
• Asynchronous Robots
• Weber Meeting Node
• Mobile Robots

Innovation

• Optimal Gathering
• Crash Faults
• Weber Meeting Node
• Asynchronous Robots
• Symmetry Breaking