Latecomers Help to Meet: Deterministic Anonymous Gathering in the Plane

TL;DR

This paper investigates the conditions under which anonymous agents with delayed start times can successfully gather in the plane, characterizing all initial configurations that allow gathering and identifying which are solvable by a universal algorithm.

Contribution

It provides a complete characterization of gatherable initial configurations and proves the existence of a universal algorithm for all good configurations, but not for bad ones.

Findings

All gatherable configurations are classified into good and bad sets.

A universal algorithm exists for gathering all good configurations of a given size.

Bad configurations, even with two agents, cannot be gathered by a single universal algorithm.

Abstract

A team of anonymous mobile agents represented by points freely moving in the plane have to gather at a single point and stop. Agents start at different points of the plane and at possibly different times chosen by the adversary. They are equipped with compasses, a common unit of distance and clocks. They execute the same deterministic algorithm and travel at speed 1. When agents are at distance at most $\epsilon$, for some positive constant $\epsilon$ unknown to them, they can exchange all information. Due to the anonymity of the agents and the symmetry of the plane, gathering is impossible, e.g., if agents start simultaneously at distances larger than $\epsilon$. However, if some agents start with a delay with respect to others, gathering may become possible. In which situations such latecomers can enable gathering? To answer this question we consider initial configurations formalized as sets of pairs $\{(p_1,t_1), (p_2,t_2),\dots , (p_n,t_n)\}$, for $n\geq 2$ where $p_i$ is the starting point of the $i$-th agent and $t_i$ is its starting time. An initial configuration is gatherable if agents starting at it can be gathered by some algorithm, even dedicated to this particular configuration. We characterize all gatherable initial configurations. Is there a universal deterministic algorithm that can gather all gatherable configurations of a given size. It turns out that the answer is no. We show that all gatherable configurations can be partitioned into two sets: bad and good configurations. We show that bad gatherable configurations (even of size 2) cannot be gathered by a common gathering algorithm, and we prove that there is a universal algorithm that gathers all good configurations of a given size.