Continuous surveillance of points by rotating floodlights

Let P and F be sets of n ≥ 2 and m ≥ 2 points in the plane, respectively, so that P∪F is in general position. We study the problem of finding the minimum angle α ∈ [2π/m, 2π] such that one can install at each point of F a stationary rotating floodlight with illumination angle α, initially oriented in a suitable direction, in such a way that, at all times, every target point of P is illuminated by at least one light. All floodlights rotate at unit speed and clockwise. We give an upper bound for the 1-dimensional problem and present results for some instances of the general problem. Specifically, we solve the problem for the case in which we have two floodlights and many points, and give an upper bound for the case in which there are many floodlights and only two target points.Ministerio de Educación y CienciaEuropean Science FoundationMinisterio de Ciencia e InnovaciónComisión Nacional de Investigación Científica y Tecnológica (Chile)Fondo Nacional de Desarrollo Científico y Tecnológico (Chile

Illuminating the x-Axis by ?-Floodlights

Given a set S of regions with piece-wise linear boundary and a positive angle α < 90°, we consider the problem of computing the locations and orientations of the minimum number of α-floodlights positioned at points in S which suffice to illuminate the entire x-axis. We show that the problem can be solved in O(n log n) time and O(n) space, where n is the number of vertices of the set S

On Barrier Graphs of Sensor Networks

The study of sensor networks begins with a model, which usually has a geometric component. This thesis focuses on networks of sensors modeled as collections of rays in the plane whose use is to detect intruders, and in particular a graph derived from this geometry, called the barrier graph of the network, which captures information about the network\u27s coverage. Every such ray-barrier sensor network corresponds to a barrier graph, but not every graph is the barrier graph of some network. We show that any barrier graph is not just tripartite, but perfect. We describe how to find networks which have certain designated graphs as their barrier graphs. We show that the size of a minimum vertex cover (in this context called the resilience) of a given graph can yield information about whether and how one can find a sensor network whose barrier graph is the given graph. Finally, we demonstrate that barrier graphs have certain strong structural properties, as a result of the geometry of ray-barrier networks, which represent progress towards a full characterization of barrier graphs