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(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 O(k2+(klog⁡nlog⁡log⁡n)ε+m) time and O(n2log⁡n+kε) space approximation algorithm, where 0<ε≤1 is the desired approximation error. For a LPGUM line and an LPGUM points set, we present two O(k2+(klog⁡nklog⁡log⁡nk)ε+mk) and O(mk2+(klog⁡nklog⁡log⁡nk)ε) time query and O((nk)2log⁡nk+kε) space approximation algorithms for the independent and dependent case, respectively.