Minimizing uncertainty through sensor placement with angle constraints

We study the problem of sensor placement in environments in which localization is a necessity, such as ad-hoc wireless sensor networks that allow the placement of a few anchors that know their location or sensor arrays that are tracking a target. In most of these situations, the quality of localization depends on the relative angle between the target and the pair of sensors observing it. In this paper, we consider placing a small number of sensors which ensure good angular α-coverage: given α in [0, π/2], for each target location t, there must be at least two sensors s₁ and s₂ such that the <(s₁ts₂) is in the interval [α, π − α]. One of the main difficulties encountered in such problems is that since the constraints depend on at least two sensors, building a solution must account for the inherent dependency between selected sensors, a feature that generic Set Cover techniques do not account for. We introduce a general framework that guarantees an angular coverage that is arbitrarily close to α for any α <= π/3 and apply it to a variety of problems to get bi-criteria approximations. When the angular coverage is required to be at least a constant fraction of α, we obtain results that are strictly better than what standard geometric Set Cover methods give. When the angular coverage is required to be at least (1 − 1/δ) · α, we obtain a O(log δ)- approximation for sensor placement with α-coverage on the plane. In the presence of additional distance or visibility constraints, the framework gives a O(log δ · log k_OPT)-approximation, where k_OPT is the size of the optimal solution. We also use our framework to give a O(log δ)-approximation that ensures (1 − 1/δ) · α-coverage and covers every target within distance 3R.