Optimal placement of mobile distance-limited devices for line routing\
Adil Erzin, Anzhela Shadrina
TL;DR
This work addresses barrier coverage by distance‑limited drones: given $m$ depots and at most $n$ drones, each drone travels a path of length at most $q$ to cover a barrier segment on $[0,L]$, with the objective of minimizing total travel length. It introduces two dynamic programming approaches: $A_1$ computes an optimal partition of the barrier into segments using the minimum total path length when the number of drones is effectively unlimited (time $O(L^2)$ after preprocessing), and $A_2$ enforces an explicit drone cap, solving a two‑parameter DP with time $O(mnL^2)$ (potentially $O(mnL)$ with improved preprocessing). The authors define per‑depot metrics $n_i(a,b)$ and $f_i(a,b)$, and show how to compute them in $O(nmL)$, enabling efficient construction of the global solution. Compared to prior work, this approach reduces preprocessing and overall complexity and guarantees an order‑preserving cover that aligns with the optimal partition. The framework enables scalable planning for limited‑resource drone barrier monitoring, with future work on noninteger endpoints and potential FPTAS developments.
Abstract
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.