A Discrete and Continuous Study of the Max-Chain-Formation Problem

TL;DR

This paper investigates the Max-Chain-Formation problem, aiming to maximize chain length of robots, analyzing strategies in discrete and continuous models, and providing bounds on convergence times with novel insights into movement strategies.

Contribution

It introduces the Max-Chain-Formation problem, analyzes discrete and continuous strategies, and derives bounds on convergence times, revealing counter-intuitive movement strategies.

Findings

Discrete model: worst-case time is O(n^2 log(n/ε))

Continuous model: optimal runtime is Θ(n)

Slowing endpoints is crucial in continuous strategies, but not in discrete ones.

Abstract

Most existing robot formation problems seek a target formation of a certain \emph{minimal} and, thus, efficient structure. Examples include the Gathering and the Chain-Formation problem. In this work, we study formation problems that try to reach a \emph{maximal} structure, supporting for example an efficient coverage in exploration scenarios. A recent example is the NASA Shapeshifter project, which describes how the robots form a relay chain along which gathered data from extraterrestrial cave explorations may be sent to a home base. As a first step towards understanding such maximization tasks, we introduce and study the Max-Chain-Formation problem, where $n$ robots are ordered along a winding, potentially self-intersecting chain and must form a connected, straight line of maximal length connecting its two endpoints. We propose and analyze strategies in a discrete and in a continuous time model. In the discrete case, we give a complete analysis if all robots are initially collinear, showing that the worst-case time to reach an $\varepsilon$-approximation is upper bounded by $\mathcal{O}(n^2 \cdot \log (n/\varepsilon))$ and lower bounded by $\Omega(n^2 \cdot~\log (1/\varepsilon))$. If one endpoint of the chain remains stationary, this result can be extended to the non-collinear case. If both endpoints move, we identify a family of instances whose runtime is unbounded. For the continuous model, we give a strategy with an optimal runtime bound of $\Theta(n)$. Avoiding an unbounded runtime similar to the discrete case relies crucially on a counter-intuitive aspect of the strategy: slowing down the endpoints while all other robots move at full speed. Surprisingly, we can show that a similar trick does not work in the discrete model.