Half-plane point retrieval queries with independent and dependent geometric uncertainties

This paper addresses a family of geometric half-plane retrieval queries of points in the plane in the presence of geometric uncertainty. The problems include exact and uncertain point sets and half-plane queries defined by an exact or uncertain line whose location uncertainties are independent or dependent and are defined by k real-valued parameters. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear approximation of geometric uncertainty that supports parametric dependencies between point locations. We present an efficient O(k²) time and space algorithm for computing the envelope of the LPGUM line that defines the half-plane query. For an exact line and an LPGUM n points set, we present an O(log⁡nk+mk) time query and O(nk) space algorithm, where m is the number of LPGUM points on or above the half-plane line. For a LPGUM line and an exact points set, we present a [Formula presented] time and [Formula presented] space approximation algorithm, where 0<ε≤1 is the desired approximation error. For a LPGUM line and an LPGUM points set, we present two [Formula presented] and [Formula presented] time query and [Formula presented] space approximation algorithms for the independent and dependent case, respectively.