Simple Robots with Minimal Sensing: From Local Visibility to Global Geometry

We consider problems of geometric exploration and selfdeployment for simple robots that can only sense the combinatorial (non-metric) features of their surroundings. Even with such a limited sensing, we show that robots can achieve complex geometric reasoning and perform many non-trivial tasks. Specifically, we show that one robot equipped with a single pebble can decide whether the workspace environment is a simply-connected polygon and, if not, it can also count the number of holes in the environment. Highlighting the subtleties of our sensing model, we show that a robot can decide whether the environment is a convex polygon, yet it cannot resolve whether a particular vertex is convex. Finally, we show that using such local and minimal sensing, a robot can compute a proper triangulation of a polygon, and that the triangulation algorithm can be implemented collaboratively by a group of m such robots, each with Θ(n/m) memory. As a corollary of the triangulation algorithm, we derive a distributed analog of the well-known Art Gallery Theorem: a group of n/3 (bounded memory) robots in our minimal sensing model can self-deploy to achieve visibility coverage of an n-vertex art gallery (polygon). This resolves an open question raised recently by Ganguli et al.