When Robots Crash: Optimal Asynchronous Gathering at Weber Meeting Nodes

We study the $\mathit{\text{optimal}}\mathit{\text{gathering}}$ problem over a finite set of designated $\mathit{\text{meeting}}\mathit{\text{nodes}}$ for $\mathit{\text{asynchronous,}}\mathit{\text{anonymous,}}$ and $\mathit{\text{oblivious}}$ 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 $\text{textsc}WeberMeetingNode$, 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 $\text{textsc}WeberMeetingNode$. We prove that, under strong multiplicity detection, optimal gathering is impossible from certain fully symmetric configurations. For all remaining configurations, our algorithm $\text{textsc}CrashTolerantWeberGathering()$ 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.