Data imprecision under λ-geometry model

Abstract

An essential assumption in most algorithms in computational geometry is that the input data is precise which is not always true in real world problems. In this paper we introduce a model – called λ-geometry model – for handling a dynamic form of imprecision which allows the precision changes in the input data of geometric problems. Also, we study two families of basic computational geometry problems under this model: axis-aligned bounding box (AABB) and proximity problems. For a set of n imprecise points where each point is modeled by a convex polygonal region with O(k) representation complexity, we propose an O(n(log n+log k+m)) time algorithm to solve the largest AABB problem, where m is the maximum number of corner defining functions when the largest AABB is defined by points on its corners. Also, we propose an O(n log n+nk+k2 log kα(k)) time algorithm for finding the smallest AABB under the assumption that the regions are disjoint. We show that the algorithms are optimal when k is O(log n). Further, we propose O(n2k log(nk) α(nk)) time algorithms for finding the closest and furthest pair problems, and present a lower bound of Ω(n2) for the closest pair problem when k is constant.