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• W1589450195 abstract "Motivated by the desire to cope with data imprecision, 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 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 (<= k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Omega(n log n) 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 Theta(n log n) steps in the worst case (under the algebraic computation tree model)." @default.
• W1589450195 created "2016-06-24" @default.
• W1589450195 creator A5089313525 @default.
• W1589450195 creator A5090438334 @default.
• W1589450195 date "2013-05-01" @default.
• W1589450195 modified "2023-09-27" @default.
• W1589450195 title "Convex hull of points lying on lines ino(nlogn)time after preprocessing" @default.
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• W1589450195 doi "https://doi.org/10.1016/j.comgeo.2012.03.004" @default.
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