A Discrete and Continuous Study of the Max-Chain-Formation Problem: Slow Down to Speed up

Abstract

Most existing robot formation problems seek a target formation of a certain 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 maximal structure, supporting for example an efficient coverage in exploration scenarios. A recent example is the NASA Shapeshifter project [24], 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 selfintersecting 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 ε-approximation is upper bounded by O(n 2 • log(n/ε)) and lower bounded by Ω(n 2 • log(1/ε)). 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 Θ(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.

The Max-Line-Formation Problem

“…Our results compare to the Max-GTM algorithm for Max-Chain-Formation (which has the same goal but considers predefined and fixed chain neighborhoods) problem as follows: our runtime of the OBLOT algorithm holds under the Ssync scheduler. For Max-GTM, only runtimes in Fsync are known [5]. Additionally, our results about Max-Line-Formation hold for every input configuration in which robots have distinct initial positions.…”

“…Very recently, the Max-Chain-Formation problem has been introduced [5]. Started in onedimensional configurations, the Max-GTM algorithm has a runtime of O(n 2 • log(n/ε)) and Ω(n 2 • log(1/ε)) rounds under the Fsync scheduler.…”

“…For two-dimensional configurations, only a convergence result is known. Additionally, for Gathering, Chain-Formation and Max-Chain-Formation, it is known that the problems can be solved optimally in a continuous time model [5,8].…”

“…Our focus lies on robots with limited visibility, i.e., each robot cannot perceive the entire swarm but only nearby robots. The terms connectivity range and viewing range are distinguished (see e.g., [5,20]). Robots are connected to all robots up to a distance equal to their connectivity range and can see all robots within their viewing range (the viewing range is at least as large as the connectivity range).…”

“…One example is the Uniform-Circle-Formation problem in which n robots are to move such that their positions form a regular polygon [11,18]. Another, very recent example and the main inspiration for this work is the Max-Chain-Formation problem [5]. The Max-Chain-Formation problem is a variant of the Chain-Formation problem.…”