Convex hull of points lying on lines in o(nlogn) time after preprocessing

  • Authors:

  • Esther Ezra;Wolfgang Mulzer

  • Affiliations:

  • Courant Institute of Mathematical Sciences, New York University, New York, NY, USA;Institut für Informatik, Freie Universität Berlin, 14195 Berlin, Germany

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2013

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Abstract

Motivated by the desire to cope with data imprecision Loffler (2009) [31], we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. 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@a(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@a(n)). The same result holds when L is a set of line segments (in general position). 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 (=