MPI-INF/SWS Research Reports 1991-2021

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A tight lower bound for the worst case of bottom-up-heapsort

Fleischer, Rudolf

April 1991, 13 pages.

Status: available - back from printing

Bottom-Up-Heapsort is a variant of Heapsort. Its worst case complexity for the number of comparisons is known to be bounded from above by ${3\over2}n\log n+O(n)$, where $n$ is the number of elements to be sorted. There is also an example of a heap which needs ${5\over4}n\log n- O(n\log\log n)$ comparisons. We show in this paper that the upper bound is asymptotical tight, i.e.~we prove for large $n$ the existence of heaps which need at least $c_n\cdot({3\over2}n\log n-O(n\log\log n))$ comparisons where $c_n=1-{1\over\log^2n}$ converges to 1. This result also proves the old conjecture that the best case for classical Heapsort needs only asymptotical $n\log n+O(n\log\log n)$ comparisons.

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  AUTHOR = {Fleischer, Rudolf},
  TITLE = {A tight lower bound for the worst case of bottom-up-heapsort},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-91-104},
  MONTH = {April},
  YEAR = {1991},
  ISSN = {0946-011X},