Convex hull of imprecise points ino(n log n)time after preprocessing
Motivated by the desire to cope with data imprecision, we study methods for preprocessing a set of line-segments (or just lines) in the plane such that whenever we are given a set of points, each of which lies on a distinct object, we can compute their convex hull more efficiently than in standard settings (that is, without preprocessing).In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(n alpha(n) log* n) expected time. If we further assume that the points are oblivious with respect to the data structure, the running time improves to O(n alpha(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-off between space and query time and an output-sensitive algorithm. We also study the dual problem where we show how to efficiently compute the (