Convex Hulls of Sets in the Plane

The convex hull of a set of points is the smallest convex set containing the points. The convex hull is a fundamental concept in mathematics and computational geometry. Other problems can be reduced to finding the convex hull – for example, halfspace intersection, Delaunay triangulation, Voronoi diagram, etc. An algorithm known as Graham scan [6] achieves an O(n log n) running time. This algorithm is optimal in the worst case. Another algorithm [7] for the same problem runs in O(nh) time, where h is the number of hull points, and this outperforms the preceding algorithm if h happens to be very small. Kirkpatrick and Seidel [8] desined later 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, Snocyink and Yap [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 of slightly more than O(n 2). Attalah [1] and Ching and Lee [3] independently presented O(n log n) runtime worst-case algorithms with O(n) space. Ching and Lee [3] also showed that this result is best possible. Devroye and Toussaint [4] and Golin, Langerman and Steiger [5] studied the case when the lines are random with certain distributions. In both models, they presented algorithms with linear expected time.