Cover Decomposability of Convex Polygons and Octants

Imagine a universe, which is basically a set of points (that may be infinite), and a collection of sensors. Each sensor has a specified covering region in the universe, i.e, a subset of the universe which it covers (monitors). Moreover, the sensors are powered by battery and they have two alternating modes of action, active and passive. In active mode a sensor covers its region and in passive mode the battery is charged and the sensor cannot work. The duration of each mode is one unit time, and once a sensor is turned on it is alternated between the two modes. The goal is to schedule the sensors in a way so that they cover the universe for all the time. One possible approach to achieve this goal is to decompose the collection of sensors into two disjoint sets so that each of these sets covers all the points of the universe. Then we can schedule the sensors in the following way. First, all the sensors of a set are turned on at the same time, say t. Then all the sensors of the other set are turned on at time t+ 1. Note that due to the one unit interval between the starting times, when the sensors of one set are charged the sensors of the other set remain active. Hence the universe gets covered for all the time. Now the fundamental problem in this approach is the decomposition of the set of sensors with such a property. Intuitively the decomposition is possible in that way if each point in the universe is covered by many sensors. In this technical report we discuss the decomposability of such sets with respect to specific types of covering region, for example intervals in R or convex polygons in R2. To define the problem formally we need some definitions at first. A family of geometric sets P in Rd is an m-fold covering of a set S if every point in S is contained in at least m sets of P . A 1-fold covering is simply called a covering. The simplest geometric sets to study are intervals in R. In Figure 1(a) a 2-fold covering of a portion of R with intervals is shown. Note that each point is covered by at least 2 intervals. Moreover, the two collections of intervals shown by dotted segments and bold segments individually covers the portion of R. Thus the whole collection of intervals is decomposable into two subcollections (dotted and bold) each of which is a