, Volume 4, Issue 1–4, pp 293–300 | Cite as

On selecting thek largest with median tests

  • Andrew Chi-Chih Yao


LetW itk(n) be the minimax complexity of selecting thek largest elements ofn numbersx 1,x 2,...,x n by pairwise comparisonsx i :x j . It is well known thatW 2(n) =n−2+ [lgn], andW k (n) = n + (k−1)lg n +O(1) for all fixed k ≥ 3. In this paper we studyW k (n), the minimax complexity of selecting thek largest, when tests of the form “Isx i the median of {x i ,x j ,x t }?” are also allowed. It is proved thatW2(n) =n−2+ [lgn], andW k (n) =n + (k−1)lg2 n +O(1) for all fixedk≥3.

Key words

Algorithm Complexity Decision tree Median test Selection 


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  1. [B]
    M. Ben-Or, Lower bounds for algebraic computation trees,Proceedings of the 15th ACM Symposium on Theory of Computing, 1983, pp. 80–86.Google Scholar
  2. [DL]
    D. Dobkin and R. J. Lipton, A lower bound of 1/2n2 on linear search tree programs for the knapsack problems,J. Comput. System Sci.,16 (1978), 413–417.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [FG]
    F. Fussenegger and H. N. Gabow, Using comparison trees to derive lower bounds for selection problems,J. Assoc. Comput. Mach.,26 (1979), 227–238.zbMATHMathSciNetGoogle Scholar
  4. [H]
    L. Hyafil, Bounds for selection,SIAM J. Comput.,5 (1976), 109–114.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Kir]
    D. G. Kirkpatrick, Topics in the complexity of combinatorial algorithms, Report TR 74, Computer Science Department, University of Toronto, 1974.Google Scholar
  6. [Kis]
    S. S. Kislytsyn, On the selection of thek-th element of an ordered set by pairwise comparisons,Sibirsk Mat. Zh.,5 (1964), 557–564.Google Scholar
  7. [Kn]
    D. E. Knuth,The Art of Computer Programming, Vol. 3, Addison-Wesley, Reading, MA, 1973.Google Scholar
  8. [PY]
    V. R. Pratt and F. F. Yao, On lower bounds for computing thei-th largest element,Proceedings of the 14th IEEE Symposium on Switching and Automata Theory, 1973, pp. 70–81.Google Scholar
  9. [Ra]
    M. Rabin, Proving simultaneous positivity of linear forms,J. Comput. System Sci.,6 (1972), 639–350.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Re]
    E. M. Reingold, Computing the maxima and the median,Proceedings of the 12th IEEE Symposium on Switching and Automata Theory, 1971, pp. 216–218.Google Scholar
  11. [SY]
    J. M. Steele and A. C. Yao, Lower bounds for algebraic decision trees,J. Algorithms,3 (1982), 1–8.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [SW]
    J. Stoer and C. Witzgall,Convexity and Optimization in Finite Dimensions I, Springer-Verlag, Berlin, 1970.Google Scholar
  13. [Y1]
    A. C. Yao, On the complexity of comparison problems using linear functions,Proceedings of the 16th Annual IEEE Symposium on Foundations of Computer Sciences, 1975, pp. 85–89.Google Scholar
  14. [Y2]
    A. C. Yao, A lower bound to finding convex hulls,J Assoc. Comput. Mach.,28 (1981), 780–787.zbMATHMathSciNetGoogle Scholar
  15. [Y3]
    F. F. Yao, On lower bounds for selection problems, Ph.D. thesis, Massachusetts Institute of Technology, 1973.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1989

Authors and Affiliations

  • Andrew Chi-Chih Yao
    • 1
  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA

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