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

Esther Ezra, Wolfgang Mulzer

نتاج البحث: نشر في مجلةمقالةمراجعة النظراء


Motivated by the desire to cope with data imprecision Löffler (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(nα(n)logn) expected time. If we further assume that the points are “oblivious” with respect to the data structure, the running time improves to O(nα(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 (⩽k)-level of n lines in the plane, each of which is incident to a distinct point (given in advance). We complement our results by Ω(nlogn) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the “standard” convex hull and sorting problems, in which the two problems require Θ(nlogn) steps in the worst case (under the algebraic computation tree model).

اللغة الأصليةالإنجليزيّة
الصفحات (من إلى)417-434
عدد الصفحات18
دوريةComputational Geometry: Theory and Applications
مستوى الصوت46
رقم الإصدار4
المعرِّفات الرقمية للأشياء
حالة النشرنُشِر - 1 مايو 2013
منشور خارجيًانعم

All Science Journal Classification (ASJC) codes

  • !!Computer Science Applications
  • !!Geometry and Topology
  • !!Control and Optimization
  • !!Computational Theory and Mathematics
  • !!Computational Mathematics


أدرس بدقة موضوعات البحث “Convex hull of points lying on lines in o(nlogn) time after preprocessing'. فهما يشكلان معًا بصمة فريدة.

قم بذكر هذا