Jaroslav Opatrny

Stefan Dobrev, Konstantinos Georgiou, Evangelos Kranakis et al.

We study a problem of online targets coverage by a drone or a sensor that is equipped with a camera or an antenna of fixed half-angle of view $α$. The targets to be monitored appear at arbitrary positions on a line barrier in an online manner. When a new target appears, the drone has to move to a location that covers the newly arrived target, as well as already existing targets. The objective is to design a coverage algorithm that optimizes the total length of the drone's trajectory. Our results are reported in terms of an algorithm's competitive ratio, i.e., the worst-case ratio (over all inputs) of its cost to that of an optimal offline algorithm. In terms of upper bounds, we present three online algorithms and prove bounds on their competitive ratios for every $α\in [0, π/2]$. The best of them, called \FA is significantly better than the other two for $π/6 < α< π/3$. In particular, for $α=π/4$, its worst case, \FA has competitive ratio $1.25$, while the other two have competitive ratio $\sqrt{2}$. Finally, we prove a lower bound on the competitive ratio of online algorithms for a drone with half-angle $α\in [0, π/4]$; this bound is a function of $α$ that achieves its maximum value at $α= π/4$ equal to $(1+\sqrt{2})/2 \approx 1.207$.

Jared Coleman, Evangelos Kranakis, Oscar Morales-Ponce et al.

In the pattern formation problem, robots in a system must self-coordinate to form a given pattern, regardless of translation, rotation, uniform-scaling, and/or reflection. In other words, a valid final configuration of the system is a formation that is \textit{similar} to the desired pattern. While there has been no shortage of research in the pattern formation problem under a variety of assumptions, models, and contexts, we consider the additional constraint that the maximum distance traveled among all robots in the system is minimum. Existing work in pattern formation and closely related problems are typically application-specific or not concerned with optimality (but rather feasibility). We show the necessary conditions any optimal solution must satisfy and present a solution for systems of three robots. Our work also led to an interesting result that has applications beyond pattern formation. Namely, a metric for comparing two triangles where a distance of $0$ indicates the triangles are similar, and $1$ indicates they are \emph{fully dissimilar}.