Filling Arbitrary Connected Areas by Silent Robots with Minimum Visibility Range

Abstract

We study the uniform dispersal problem (also called the filling problem) in arbitrary connected areas. In the filling problem robots are injected one-by-one at $k \ge 1$ Doors into an unknown area, subdivided into cells. The goal is to cover the area, i.e. each cell must be occupied by a robot. The robots are homogeneous, anonymous, autonomous, have limited visibility radius, limited persistent memory, and silent, i.e. do not use explicit communication. A fundamental question is how ‘weak’ those robots can be in terms of hardware requirements and still be able to solve the problem, which was initiated by Barrameda et al. [4]. In our previous paper [11] we presented an algorithm which solves the filling problem for orthogonal areas with O(1) bits of persistent memory, 1 hop visibility range and without explicit communication. The algorithm utilized the timing of movements and had O(n) runtime, where n is the number of cells in the area. In this paper, we generalize the problem for silent robots for an arbitrary connected area represented by a graph, while maintaining the 1 hop visibility range. The algorithm is collision-free, it terminates in $O(k \cdot \varDelta \cdot n)$ rounds, and requires $O(\varDelta \cdot \log k)$ bits of persistent memory, where $\varDelta$ is the maximum degree of the graph.