[Paper Review] Modem Illumination of Monotone Polygons

Oswin Aichholzer, Ruy Fabila Monroy+4 more|RECERCAT (Consorci de Serveis Universitaris de Catalunya)|Mar 17, 2015

Manufacturing Process and OptimizationEngineering19 citations

TL;DR

This paper introduces the k-modem illumination problem, modeling wireless signals that penetrate up to k walls in a polygonal environment. It establishes tight upper and lower bounds on the minimum number of k-modems needed to illuminate monotone and monotone orthogonal polygons, showing that ⌈(n−2)/(2k+3)⌉ k-modems suffice and are necessary for monotone polygons, with refined bounds depending on k's parity for orthogonal cases.

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 $\big\lceil \frac{n-2}{2k+3} \big ceil$ $k$-modems. In addition, we exhibit examples of monotone polygons requiring at least $\lceil \frac {n-2} {2k+3} ceil$ $k$-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 $\big\lceil \frac{n-2}{2k+4} \big ceil$ $k$-modems, while for odd $k\geq3$, $\big\lceil \frac{n-2}{2k+6} \big ceil$ $k$-modems are always sufficient. Further, by presenting according examples of monotone orthogonal polygons, we show that both bounds are tight.

Motivation & Objective

• To model wireless signal propagation in buildings using k-modems that penetrate up to k walls.
• To determine the minimum number of k-modems required to illuminate any monotone polygon with n vertices.
• To extend the analysis to monotone orthogonal polygons, accounting for geometric constraints and k's parity.
• To prove tight upper and lower bounds for both monotone and monotone orthogonal polygon classes.
• To establish algorithmic foundations for optimal placement of wireless devices in polygonal environments.

Proposed method

• Introduces the k-modem model, where a point in a polygon is illuminated if the line segment to the modem crosses at most k walls.
• Applies a splitting lemma to divide monotone polygons into smaller x-monotone subpolygons, enabling recursive decomposition.
• Uses structural properties of monotone and orthogonal polygons to bound the number of vertices in subpolygons after splitting.
• Employs case analysis based on edge configurations (e.g., horizontal edges and splitting lines) to ensure subpolygons remain x-monotone and orthogonal.
• Applies lemmas on stair-end polygons and subpolygon illumination to prove that a single k-modem suffices per subpolygon.
• Combines recursive decomposition with subpolygon illumination to derive global bounds using ceiling functions.

Experimental results

Research questions

• RQ1 What is the minimum number of k-modems required to illuminate any monotone polygon with n vertices?
• RQ2 How does the number of required k-modems change for monotone orthogonal polygons, particularly based on the parity of k?
• RQ3 Can tight upper and lower bounds be established for k-modem placement in monotone and monotone orthogonal polygons?
• RQ4 Is the proposed bound of ⌈(n−2)/(2k+3)⌉ k-modems for monotone polygons both sufficient and necessary in the worst case?
• RQ5 How does orthogonality and k's parity affect the illumination complexity compared to general monotone polygons?

Key findings

• For any monotone polygon with n vertices, ⌈(n−2)/(2k+3)⌉ k-modems are both sufficient and necessary in the worst case.
• There exist monotone polygons that require at least ⌈(n−2)/(2k+3)⌉ k-modems, proving the bound is tight.
• For monotone orthogonal polygons, ⌈(n−2)/(2k+4)⌉ k-modems are sufficient and necessary when k=1 or k is even.
• For odd k≥3, ⌈(n−2)/(2k+6)⌉ k-modems are sufficient and necessary to illuminate any monotone orthogonal n-gon.
• The bounds for k=1 are identical to those for even k, due to geometric constraints and small constant values.
• The paper constructs explicit families of monotone and monotone orthogonal polygons that achieve the lower bounds, confirming tightness.