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- W4287646078 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." @default.
- W4287646078 created "2022-07-25" @default.
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- W4287646078 date "2020-10-05" @default.
- W4287646078 modified "2023-09-23" @default.
- W4287646078 title "A Discrete and Continuous Study of the Max-Chain-Formation Problem" @default.
- W4287646078 doi "https://doi.org/10.48550/arxiv.2010.02043" @default.
- W4287646078 hasPublicationYear "2020" @default.
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