Semi-Uniform Deployment of Mobile Robots in Perfect $\ell$ -ary Trees

Abstract

In this paper, we consider the problem of semi-uniform deployment for mobile robots in perfect $\ell$-ary trees, where every intermediate node has $\ell$ children, and all leaf nodes have the same depth. This problem requires robots to spread in the tree so that, for some positive integer $d$ and some fixed integer $s(0\leq s\leq d-1)$, each node of depth $s+dj\ (j\geq 0)$ is occupied by a robot. In other words, after semi-uniform deployment is achieved, nodes of depth $s, s+d, s+2d, \ldots$ are occupied by a robot. Robots have an infinite visibility range but are opaque, that is, robot $r_{i}$ cannot observe some robot $r_{j}$ if there exists another robot $r_{k}$ in the path between $r_{i}$ and $r_{j}$. In addition, each robot can emit a light color visible to itself and other robots, taken from a set of $\kappa$ colors, at each time step. Then, we clarify the relationship between the number of available light colors and the solvability of the semi-uniform deployment problem. First, we consider robots with the minimum number of available light colors, that is, robots with $\kappa =1$ (in this case, robots are oblivious). In this setting, we show that there is no collision-free algorithm to solve the semi-uniform deployment problem with explicit termination. Next, we relax the number of available light colors, that is, we consider robots with $\kappa=2$. In this setting, we propose a collision-free algorithm that can solve the problem with explicit termination. Thus, our algorithm is optimal with respect to the number of light colors. In addition, to the best of our knowledge, this paper is the first to report research considering (a variant of) uniform deployment in graphs other than rings or grids.