Modem illumination of monotone polygons

Abstract

We study a generalization of the classical problem of the illumination of polygons. Instead of modeling a light source we model a wireless device whose radio signal can penetrate a given number k of walls. We call these objects k-modems and study the minimum number of k-modems sufficient and sometimes necessary to illuminate monotone and monotone orthogonal polygons. We show that every monotone polygon with n vertices can be illuminated with n−2 2k+3 k-modems. In addition, we exhibit examples of monotone polygons requiring at least n−2 2k+3 k-modems to be illuminated. For monotone orthogonal polygons with n vertices we show that for k = 1 and for even k, every such polygon can be illuminated with n−2 2k+4 k-modems, while for odd k ≥ 3, n−2 2k+6 k-modems are always sufficient. Further, by presenting according examples of monotone orthogonal polygons, we show that both bounds are tight.

Combinatorics and complexity of guarding polygons with edge and point 2-transmitters

“…In the case of orthogonal polygons, the bound becomes n 4 , as shown by Kahn et al [13]. For k-transmitters, Aichholzer et al [1] showed n 2k k-transmitters are always sufficient and n 2k+4 k-transmitters are sometimes necessary to cover a monotone n-gon 1 ; for monotone orthogonal polygons they gave a tight bound of n−2 2k+4 k-transmitters, for k even and k = 1. Aichholzer et al [2] improved the bounds on monotone polygons to a tight value of n−2 2k+3 .…”

“…Aichholzer et al [1] first formalized this problem by considering k-modems (k-transmitters), devices whose wireless signal can pass through at most k walls. Since 2010, little progress has been made on the problem of k-transmitters, or even the problem of 2-transmitters, despite reaching a wide audience as the topic of a computational geometry column by Joseph O'Rourke [18] in the SIGACT News in 2012.…”

A time–space trade-off for computing the k-visibility region of a point in a polygon

“…Since 2009, this variant of visibility has been explored more widely due to its relevance in wireless networks. In particular, it models the coverage areas of wireless devices whose radio signals can penetrate up to k walls [2,14]. This makes the problem particularly interesting for the limited workspace model, since these wireless devices are typically equipped with only a small amount of memory for computational tasks and may need to determine their coverage region using the few resources at their disposal.…”

“…We believe that our ideas are also applicable for this notion and lead to an improvement of their result. 2 Related work. The optimal classic algorithm for computing the 0-visibility region needs O(n) time and O(n) space [19].…”

“…This takes 1 For k = n − 1, the whole polygon is k-visible from q, so there is no reason to consider k > n − 1. 2 The algorithm of Bajuelos et al [4] essentially first computes a complete arrangement of quadratic size that encodes the whole visibility information, and then extracts the k-visible region from this arrangement. Our algorithms, on the other hand, use a plane sweep so that only the relevant parts of this arrangement are considered.…”