Optimal placement of mobile distance-limited devices for line routing\

A segment (barrier) is specified on the plane, as well as depots, where the mobile devices (drones) can be placed. Each drone departs from its depot to the barrier, moves along the barrier and returns to its depot, traveling a path of a limited length. The part of the barrier along which the drone moved is \emph{covered} by this sensor. It is required to place a limited quantity of drones in the depots and determine the trajectory of each drone in such a way that the barrier is covered, and the total length of the paths traveled by the drones is minimal. Previously, this problem was considered for an unlimited number of drones. If each drone covers a segment of length at least 1, then the time complexity of the proposed algorithm was $O(mL^3)$, where $m$ is the number of depots and $L$ is the length of the barrier. In this paper, we generalize the problem by introducing an upper bound $n$ on the number of drones, and propose a new algorithm with time complexity equals $O(mnL^2)$. Since each drone covers a segment of length at least 1, then $n\leq L$ and $O(mnL^2)\leq O(mL^3)$. Assuming an unlimited number of drones, as investigated in our prior work, we present an $O(mL^2)$-time algorithm, achieving an $L$-fold reduction compared to previous methods. Here, the algorithm has a time complexity that equals $O(L^2)$, and the most time-consuming is preprocessing.