Improved Linear Expected-Time Algorithms for Computing Maxima
The problem of finding the maxima of a point set plays a fundamental role in computational geometry. Based on the idea of the certificates of exclusion, two algorithms are presented to solve the maxima problem under the assumption that N points are chosen from a d-dimensional hypercube uniformly and each component of a point is independent of all other components. The first algorithm runs in O(N) expected time and finds the maxima using dN + dln N + d2N 1 − 1/d (lnN)1/d + O(dN1 − 1/d) expected scalar comparisons. The experiments show the second algorithm has a better expected running time than the first algorithm while a tight upper bound of the expected running time is not obtained. A third maxima-finding algorithm is presented for N points with a d-dimensional component independence distribution, which runs in O(N) expected time and uses 2dN + O(ln N(ln(ln N))) + d 2 N 1 − 1/d (lnN)1/d + O(dN1 − 1/d) expected scalar comparisons. The substantial reduction of the expected running time of all three algorithms, compared with some known linear expected-time algorithms, has been attributed to the fact that a better certificate of exclusion has been chosen and more non-maximal points have been identified and discarded.
KeywordsComputational Geometry Recursive Partition Scalar Comparison Unit Hypercube Dynamic Maintenance
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