  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 Omega(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.
