A note on the worst case of heapsort

A Tight Bound on the Worst-Case Number of Comparisons for Floyd’s Heap Construction Algorithm

“…Kruskal et al [6] showed that 2n − 2 log(n + 1) is a tight bound on the worstcase number of comparisons, if n = 2 k − 1, where k is a positive integer, and, to our knowledge, this is the only value of n for which a tight upper bound has been reported in the literature. Schaffer [7] showed that n − log(n + 1) + λ(n), where λ(n) is the number of zeros in the binary representation of n, is the sum of heights of sub-trees rooted at internal nodes of a complete binary tree, see also [3] for an interesting geometric approach to the same problem.…”

“…Despite the overwhelming attention received by the computer community in the more than 45 years of its life, a tight bound on the worst-case number of comparisons holding for all values of n, is, to our knowledge, still unknown. Kruskal et al [6] showed that 2n − 2 log(n + 1) is a tight bound on the worstcase number of comparisons, if n = 2 k − 1, where k is a positive integer, and, to our knowledge, this is the only value of n for which a tight upper bound has been reported in the literature.…”