Fast Linear Expected-Time Algorithms for Computing Maxima and Convex Hulls

Jon Louis Bentley, Kenneth L. Clarkson, David B. Levine

Abstract: This paper examines the expected complexity of boundary problems on a set ofN points inK-space. We assume that the points are chosen from a probability distribution in which each component of a point is chosen independently of all other components. We present an algorithm to find the maximal points usingKN + O (N1−1/K log1/K N) expected scalar comparisons, for fixedK≥ 2. A lower bound shows that the algorithm is optimal in the leading term. We describe a simple maxima algorithm that is easy to code, and present experimental evidence that it has similar running time. For fixedK ≥ 2, an algorithm computes the convex hull of the set in 2KN + O(N1−1/K log1/KN) expected scalar comparisons. The history of the algorithms exhibits interesting interactions among consulting, algorithm design, data analysis, and mathematical analysis of algorithms.