An upper bound on the k-modem illumination problem

TL;DR

This paper establishes an upper bound of O(n/k) on the number of k-modems needed to illuminate any polygon with n vertices, improving understanding of wireless illumination constraints.

Contribution

The paper proves a new upper bound of O(n/k) for the k-modem illumination problem and provides an efficient algorithm to achieve this bound.

Findings

Upper bound of O(n/k) for general polygons.

Tighter bound of 6n/k + 1 for orthogonal polygons.

An O(n log n) algorithm to compute the illumination set.

Abstract

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.