Optimal Gathering over Weber Meeting Nodes in Infinite Grid

TL;DR

This paper presents an optimal deterministic algorithm for robot gathering on an infinite grid that minimizes total movement, addressing the problem under asynchronous conditions and characterizing solvable configurations.

Contribution

It introduces a new optimal gathering algorithm for robots on an infinite grid with meeting nodes, considering asynchronous operation and move minimization.

Findings

The algorithm is proven optimal in total move count.

It characterizes initial configurations where gathering is impossible.

The solution works under asynchronous robot models.

Abstract

The gathering over meeting nodes problem requires the robots to gather at one of the pre-defined meeting nodes. This paper investigates the problem with respect to the objective function that minimizes the total number of moves made by all the robots. In other words, the sum of the distances traveled by all the robots is minimized while accomplishing the gathering task. The robots are deployed on the nodes of an anonymous two-dimensional infinite grid which has a subset of nodes marked as meeting nodes. The robots do not agree on a global coordinate system and operate under an asynchronous scheduler. A deterministic distributed algorithm has been proposed to solve the problem for all those solvable configurations, and the initial configurations for which the problem is unsolvable have been characterized. The proposed gathering algorithm is optimal with respect to the total number of moves performed by all the robots in order to finalize the gathering.