Triangle Evacuation of 2 Agents in the Wireless Model

TL;DR

This paper extends the study of evacuation algorithms to arbitrary non-obtuse triangles, analyzing optimal strategies and bounds for two agents starting on the perimeter, building on prior work on symmetric domains.

Contribution

It introduces new algorithmic problems for evacuation in non-symmetric triangles and provides tight bounds, extending previous results from equilateral triangles to more general shapes.

Findings

Optimal algorithms match lower bounds for non-obtuse triangles.

Extended analysis from equilateral to arbitrary non-obtuse triangles.

Provided tight bounds for multiple starting configurations.

Abstract

The input to the \emph{Triangle Evacuation} problem is a triangle $ABC$. Given a starting point $S$ on the perimeter of the triangle, a feasible solution to the problem consists of two unit-speed trajectories of mobile agents that eventually visit every point on the perimeter of $ABC$. The cost of a feasible solution (evacuation cost) is defined as the supremum over all points $T$ of the time it takes that $T$ is visited for the first time by an agent plus the distance of $T$ to the other agent at that time. Similar evacuation type problems are well studied in the literature covering the unit circle, the $\ell_p$ unit circle for $p\geq 1$, the square, and the equilateral triangle. We extend this line of research to arbitrary non-obtuse triangles. Motivated by the lack of symmetry of our search domain, we introduce 4 different algorithmic problems arising by letting the starting edge and/or the starting point $S$ on that edge to be chosen either by the algorithm or the adversary. To that end, we provide a tight analysis for the algorithm that has been proved to be optimal for the previously studied search domains, as well as we provide lower bounds for each of the problems. Both our upper and lower bounds match and extend naturally the previously known results that were established only for equilateral triangles.