Worst-Case Analysis of Heapsort, Exactly

Abstract

Abstract This paper completes analysis of the worst-case running time of $ {\tt{Heapsort}}$ measured by the number of comparisons of keys performed while sorting. A derivation of an exact formula for the maximum number of comparisons of keys performed by $ {\tt{Heapsort}} $ on any array of size $ N \geq 2 $ is presented. It is equal to $$ \begin{align}& \nonumber 2(N-1)\lceil \lg N \rceil - 2 ^{\lceil \lg N \rceil +1} - 2 s_2(N) - e_2(N) + \min (\lceil \lg N \rceil, 3) + 5 + c, \end{align} $$ where $s_2(N)$ is the sum of digits of the binary representation of $N$, $e_2(N)$ is the exponent of $2$ in the $N$’s prime factorization and $ c $ is a binary function on the set of positive integers defined by $$ \begin{align}& \nonumber c = \left\{ \begin{array}{@{}ll} 1 \text{ if} \; N \leq 2 ^{\lceil \lg N \rceil} - 4 \\[4pt] 0 \text{ otherwise}. \end{array} \right. \end{align} $$ The above formula allows for deciding, in $ O(N \log N) $ time, if any given $ N $-element array is a worst-case array for $ {\tt{Heapsort}} $. Its proof yields an algorithm for construction, in $ O(N \log N) $ time, of worst-case arrays of arbitrary sizes $ N \geq 2 $ for $ {\tt{Heapsort}} $.