An upper bound on the k-modem illumination problem

A variation on the classical polygon illumination problem was introduced in [Aichholzer et. al. EuroCG’09]. In this variant light sources are replaced by wireless devices called k-modems, which can penetrate a fixed number k, of “walls”. A point in the interior of a polygon is “illuminated” by a k-modem if the line segment joining them intersects at most k edges of the polygon. It is easy to construct polygons of n vertices where the number of k-modems required to illuminate all interior points is Ω(n/k). However, no non-trivial upper bound is known. In this paper we prove that the number of k-modems required to illuminate any polygon of n vertices is at most O(n/k). For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of 6n/k+1. Moreover, we present an O(n log n) time algorithm to achieve this bound.