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Fast linear expected-time algorithms for computing maxima and convex hulls. (English) Zbl 0766.68132
Summary: This paper examines the expected complexity of boundary problems on a set of \(N\) points in \(K\)-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 using \(KN+O(n^{1-1/K}\log^{1/K}N)\) expected scalar comparisons, for fixed \(K\geq 2\). 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 fixed \(K\geq 2\), an algorithm computes the convex hull of the set in \(2KN+O(N^{1-1/K}N)\) expected scalar comparisons. The history of the algorithms exhibits interesting interactions among consulting, algorithm design, data analysis, and mathematical analysis of algorithms.

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
68Q25 Analysis of algorithms and problem complexity
Full Text: DOI
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