Algorithms for art gallery illumination

Abstract

The art gallery problem (AGP) is one of the classical problems in computational geometry. It asks for the minimum number of guards required to achieve visibility coverage of a given polygon. The AGP is well-known to be NP-hard even in restricted cases. In this paper, we consider the AGP with fading (AGPF): A polygonal region is to be illuminated with light sources such that every point is illuminated with at least a global threshold, light intensity decreases over distance, and we seek to minimize the total energy consumption. Choosing fading exponents of zero, one, and two are equivalent to the AGP, laser scanner applications, and natural light, respectively. We present complexity results as well as a negative solvability result. Still, we propose two practical algorithms for AGPF with fixed light positions (e.g. Glob Optim (2017) 68:23-45 vertex guards) independent of the fading exponent, which we demonstrate to work well in practice. One is based on a discrete approximation, the other on non-linear programming by means of simplex-partitioning strategies. The former approach yields a fully polynomial-time approximation scheme for the AGPF with fixed light positions. The latter approach obtains better results in our experimental evaluation. B

Engineering Art Galleries

“…instead of (2). Two algorithms for vertex guards were proposed and tested [29], based on the BS-2013 implementation. The first approximates with a step function, and uses updated primal and dual separation routines that operate on overlays of visibility polygons and circular arcs, resulting in an FPTAS for the fractional AGP(V, P ).…”

On Voronoi visibility maps of 1.5D terrains with multiple viewpoints

“…These problems are motivated by the fact that, even though many visibility problems assume an infinite range of visibility, the intensity of light, signals and other phenomena modelled with viewpoints decreases over distance in realistic environments. In this spirit, the problem of illuminating a polygonal area with the minimum total energy was introduced by O'Rourke [16], and studied in [7,8]. We consider a related problem on terrains, namely, computing the minimum value r * such that, if the viewpoints can only see objects within distance r * , the obtained visibility map is the same as Vis(T , P).…”

The Continuous 1.5D Terrain Guarding Problem: Discretization, Optimal Solutions, and PTAS

“…The authors would like to revive a motivation stemming from research regarding algorithms solving the AGP [10,12,16,17,34] already mentioned in [1]: An application of the AGP is the placement of sensors or communication devices w.r.t. obstacles, for example placing laser scanners in production facilities to acquire a precise mapping of the facility [16,34].…”

“…The authors would like to revive a motivation stemming from research regarding algorithms solving the AGP [10,12,16,17,34] already mentioned in [1]: An application of the AGP is the placement of sensors or communication devices w.r.t. obstacles, for example placing laser scanners in production facilities to acquire a precise mapping of the facility [16,34]. While the AGP properly models most indoor environments it cannot capture many outdoor scenarios, like placing cell phone towers in an urban environment, because it does not take height information into account.…”