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Improved Linear Expected-Time Algorithms for Computing Maxima

  • H. K. Dai
  • X. W. Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

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.

Keywords

Computational Geometry Recursive Partition Scalar Comparison Unit Hypercube Dynamic Maintenance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • H. K. Dai
    • 1
  • X. W. Zhang
    • 1
  1. 1.Computer Science DepartmentOklahoma State UniversityStillwaterU.S.A.

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