The Max-Line-Formation Problem

We consider n robots with limited visibility: each robot can observe other only up to a constant distance denoted as the viewing range. The operate in discrete rounds that are either fully synchronous (FSync) or semi-synchronized (SSync). Most previously studied formation problems this setting seek bring closer together (e.g., Gathering Chain-Formation). In work, we introduce Max-Line-Formation problem, which has contrary goal: arrange on straight line of maximal length. First, prove problem is impossible solve by sized circular impossibility holds under comparably strong assumptions: agree both axes their local coordinate systems FSync. On positive side, show solvable square range, i.e., lie within constant-sized centered at position. case, need one axis systems. derive two algorithms: first algorithm considers oblivious and converges optimal configuration time $\mathcal{O}(n^2 \cdot \log (n/\varepsilon))$ SSync scheduler. makes use locally visible lights (LUMI). It designed for FSync scheduler exactly $\Theta(n)$. Afterward, algorithmic analysis techniques also be applied Chain-Formation problem: an reduced range give new improved runtime bounds problem.