Towards Gathering Robots with Limited View in Linear Time: The Closed Chain Case

TL;DR

This paper presents the first linear-time gathering algorithm for a closed chain of anonymous robots on a grid, overcoming the previous quadratic runtime lower bound in the fully synchronous model.

Contribution

It introduces a novel sub-quadratic, specifically linear, gathering algorithm for closed chains of robots with limited local view in the FSYNC model.

Findings

Achieves linear time gathering for closed chains on a grid.

Breaks the quadratic lower bound for this problem.

Operates under limited memory and local knowledge constraints.

Abstract

In the gathering problem, n autonomous robots have to meet on a single point. We consider the gathering of a closed chain of point-shaped, anonymous robots on a grid. The robots only have local knowledge about a constant number of neighboring robots along the chain in both directions. Actions are performed in the fully synchronous time model FSYNC. Every robot has a limited memory that may contain one timestamp of the global clock, also visible to its direct neighbors. In this synchronous time model, there is no limited view gathering algorithm known to perform better than in quadratic runtime. The configurations that show the quadratic lower bound are closed chains. In this paper, we present the first sub-quadratic---in fact linear time---gathering algorithm for closed chains on a grid.