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

Abstract

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 $O\left(k^2\right)$ 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 $O\left(nk\right)$ 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 $O(k^2+\frac{(k\log n\log \log n)}{\epsilon }+m)$ time and $O(\frac{n^2}{\log n}+\frac{k}{\epsilon })$ space approximation algorithm, where $0<\epsilon \le 1$ is the desired approximation error. For a LPGUM line and an LPGUM points set, we present two $O(k^2+\frac{(k\log nk\log \log nk)}{\epsilon }+mk)$ and $O(m{k}^2+\frac{(k\log nk\log \log nk)}{\epsilon })$ time query and $O(\frac{{\left(nk\right)}^2}{\log nk}+\frac{k}{\epsilon })$ space approximation algorithms for the independent and dependent case, respectively.