Point set distance and orthogonal range problems with dependent geometric uncertainties

Classical computational geometry algorithms handle geometric constructs whose shapes and locations are exact. However, many real-world applications require modeling and computing with geometric uncertainties, which are often coupled and mutually dependent. In this paper we address the relative position of points, point set distance problems, and orthogonal range queries in the plane in the presence of geometric uncertainty. The uncertainty can be in the locations of the points, in the query range, or both, and is possibly coupled. Point coordinates and range uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), a general and computationally efficient worst-case, first-order linear approximation of geometric uncertainty that supports dependence among uncertainties. We present efficient algorithms for relative points orientation, minimum and maximum pairwise distance, closest pair, diameter, and efficient algorithms for uncertain range queries: uncertain range/nominal points, nominal range/uncertain points, uncertain range/uncertain points, with independent/dependent uncertainties. In most cases, the added complexity is sub-quadratic in the number of parameters and points, with higher complexities for dependent point uncertainties.