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