The Surplus Parking Gathering Problem in Infinite Grids

2026-07-07

AI Summary

This work formally defines, for the first time, the surplus parking aggregation problem on infinite grids: when the total number of robots exceeds the combined capacity of designated parking nodes, how to saturate each parking node exactly while gathering the excess robots collision-free into a single, uniquely determined common node. To address this problem, we propose the first deterministic distributed algorithm that is provably correct and terminates in finite time under the asynchronous (ASYNC) model. Leveraging global visibility and strong multiplicity detection, the algorithm coordinates multi-stage movements to ensure conflict-free navigation. It achieves a movement complexity of $O(n(a+b) + n^2)$ and matches a proven theoretical lower bound of $\Omega(n(a+b))$, demonstrating near-optimal performance in the worst case.

Abstract

In this paper, we introduce the \emph{Surplus Parking Gathering Problem} ($\mathcal{SPG}$), a new coordination problem for robots deployed on an infinite grid. The input consists of a set of designated parking nodes, each associated with a prescribed capacity, while the total number of robots exceeds the total parking capacity. The objective is to saturate every parking node exactly according to its capacity while gathering all remaining surplus robots at a common grid node that is not specified a priori. The robots are assumed to be autonomous, anonymous, oblivious, identical, disoriented, and homogeneous. We consider the asynchronous (\textsc{async}) model with global visibility and global strong multiplicity detection. We first establish necessary conditions for the solvability of $\mathcal{SPG}$ by characterizing the initial configurations that admit no deterministic distributed algorithm. For all the remaining solvable configurations, we present a deterministic distributed algorithm that correctly solves the problem. The proposed algorithm proceeds in several phases and avoids collisions throughout its execution. We prove that the algorithm terminates in finite time and, upon termination, every parking node is saturated according to its prescribed capacity while all surplus robots are gathered at a uniquely determined gathering node. We further analyze the move complexity of the proposed algorithm, obtaining an $O(n(a+b)+n^2)$ upper bound together with an $Ω(n(a+b))$ worst-case lower bound for the $\mathcal{SPG}$ problem.

Problem

• Surplus Parking Gathering Problem
• infinite grids
• robot coordination
• parking capacity
• surplus robots

Innovation

• Surplus Parking Gathering
• deterministic distributed algorithm
• asynchronous robot coordination
• move complexity
• infinite grid