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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.

##### MSC:
 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 68Q25 Analysis of algorithms and problem complexity
##### Keywords:
maximal points; maxima algorithm; convex hull
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##### References:
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