Semi‐uniform deployment of mobile robots in perfect ℓ$$ \ell $$‐ary trees

Abstract

SummaryIn this paper, we consider the problem of semi‐uniform deployment for mobile robots in perfect ‐ary trees. This problem requires robots to spread in the tree so that, for some positive integer and some fixed integer , each node of depth is occupied by a robot. Robots have an infinite visibility range but are opaque, and each robot can emit a light color visible to itself and other robots, taken from a set of colors, at each time step. Then, we clarify the solvability of the semi‐uniform deployment problem, focusing on the number of available light colors. First, we consider robots with . In this setting, we show that there is no collision‐free algorithm to solve the problem. Next, relax the number of available light colors, that is, we consider robots with . In this setting, we propose a collision‐free algorithm that can solve the problem. From these results, we can show that the semi‐uniform deployment problem can be solved when , and our proposed algorithm is optimal with respect to the number of used light colors (i.e., 2).