Convex Hull of Imprecise Points in o(n \log{n}) Time after Preprocessing

Motivated by the desire to cope with data imprecision [8], we study methods for preprocessing a set of planar regions such that whenever we are given a set of points, each of which lies on a distinct region, we can compute a specified structure on these points 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(nα(n) log∗ n) expected time. The analysis applies almost verbatim when L is a set of line-segments, and yields the same asymptotic bounds.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Esther Ezra, Wolfgang Mulzer
• Keywords: Convex Hull, CORR 2011, Data Imprecision, Education, Planar Regions