Convex hull for intersections of random lines

Abstract

Numerous problems can be reduced to finding the convex hull of a set of points – halfspace intersection, Delaunay triangulation, etc. An algorithm for finding the convex hull in the plane, known as Graham scan [5], achieves an O(n log n) running time. This algorithm is optimal in the worst case. Another algorithm [6] for the same problem runs in O(nh) time, where h is the number of hull points, and outperforms the preceding algorithm if h happens to be very small. Kirkpatrick and Seidel [7] designed an algorithm with O(n log h) runtime, which is always at least as good as the better of the above two algorithms. A simplification of their algorithm has been recently reported by Chan [2]. For some special sets of points, it is possible to improve the results above. We concentrate on the case when the set consists of all intersection points of n lines. A straightforward application of the algorithms above leads to a runtime O(n log n). Attalah [1] and Ching and Lee [3] independently presented O(n log n) runtime worst-case algorithms withO(n) space. Ching and Lee [3] also showed that this result is best possible. Devroye and Toussaint [4] and Golin, Langerman and Steiger [8] studied the case when the lines are random with certain distributions. (Note that our model is different from the model of [8].) In both models, they presented algorithms with linear expected time. Let us concentrate on the model of Devroye and Toussaint. It is convenient to use the representation of lines by points. A line not passing through the origin is uniquely determined by its intersection point with the line perpendicular to it from the origin. It is often useful to define a mechanism for selecting random lines via a mechanism for a random selection of the corresponding intersection points. In the model of Devroye and Toussaint [4] all lines are independent identically distributed. The polar coordinates of the corresponding points are selected as follows. The distance from the origin is distributed according to some distribution lawR (required to have a finite mean) and the angle is distributed uniformly in [0, 2π); the distance and the angle are independent. As mentioned earlier, their algorithm works in linear expected time. The linearity follows from a result they claim for the set of outer layer points. Here, given a set S, the outer layer of S consists of those points P ∈ S such that at least one quadrant around P does not contain any other point of S. Clearly, any point in the convex hull belongs also to the outer layer. Their theorem asserts that the expected number of outer layer points is bounded above by some constant. In fact, Devroye and Toussaint proved that, given a distribution R, there exists a constant C such that, denoting byOn the number of outer layer points arising from n lines, we haveE(On) ≤ C for sufficiently large n. If one could find a constant C and an N0 such that E(On) ≤ C for every n ≥ N0, independently of R, the problem would be completely solved. However, we show by means of counter-examples that no such C and N0 exist. As indicated above, the result of Devroye and Toussaint [4] regarding the expected number of layer points was proved under the assumption that the distribution R has a finite mean. Here we construct a

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Cite this paper

@inproceedings{Berend2005ConvexHF, title={Convex hull for intersections of random lines}, author={Daniel Berend and Vladimir Braverman}, year={2005} }