Efficient Sensor Placement for Surveillance Problems

We study the problem of covering a two-dimensional spatial region P , cluttered with occluders, by sensors. A sensor placed at a location p covers a point x in P if x lies within sensing radius r from p and x is visible from p , i.e., the segment px does not intersect any occluder. The goal is to compute a placement of the minimum number of sensors that cover P . We propose a landmark-based approach for covering P . Suppose P has *** holes, and it can be covered by h sensors. Given a small parameter *** 0, let *** : = *** (h ,*** ) = (h /*** )(1 + ln (1 + *** )). We prove that one can compute a set L of O (*** log*** log(1/*** )) landmarks so that if a set S of sensors covers L , then S covers at least (1 *** *** )-fraction of P . It is surprising that so few landmarks are needed, and that the number of landmarks depends only on h , and does not directly depend on the number of vertices in P . We then present efficient randomized algorithms, based on the greedy approach, that, with high probability, compute $O(\tilde{h}\log \lambda)$ sensor locations to cover L ; here $\tilde{h} \le h$ is the number sensors needed to cover L . We propose various extensions of our approach, including: (i) a weight function over P is given and S should cover at least (1 *** *** ) of the weighted area of P , and (ii) each point of P is covered by at least t sensors, for a given parameter t *** 1.