Reducing the Complexity of the Sensor-Target Coverage Problem Through Point and Set Classification

The problem of covering random points in a plane with sets of a given shape has several practical applications in communications and operations research. One especially prominent application is the coverage of randomly-located points of interest by randomly-located sensors in a wireless sensor network. In this article we consider the situation of a large area containing randomly placed points (representing points of interest), as well a number of randomly-placed disks of equal radius in the same region (representing individual sensors' coverage areas). The problem of finding the smallest possible set of disks that cover the given points is known to be NP-complete. We show that the computational complexity may be reduced by classifying the disks into several definite classes that can be characterized as necessary, excludable, or indeterminate. The problem may then be reduced to considering only the indeterminate sets and the points that they cover. In addition, indeterminate sets and the points that they cover may be divided into disjoint ``islands'' that can be solved separately. Hence the actual complexity is determined by the number of points and sets in the largest island. We run a number of simulations to show how the proportion of sets and points of various types depend on two basic scale-invariant parameters related to point and set density. We show that enormous reductions in complexity can be achieved even in situations where point and set density is relatively high.