Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing

TL;DR

This paper introduces a preprocessing method for line sets that allows efficient convex hull computation of points on these lines, achieving faster query times than traditional methods, with extensions and lower bounds provided.

Contribution

It presents a novel preprocessing technique for line sets enabling faster convex hull queries for points on those lines, surpassing standard sorting complexities.

Findings

Convex hull of points on lines can be computed in O(n alpha(n) log* n) expected time after preprocessing.

Preprocessing requires quadratic time and space to enable fast queries.

The problem is shown to be easier than sorting under the algebraic computation tree model.

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).