heapsort

The heapsort algorithm is an elegant application of the heap data structure to the sorting problem. It consists of building a heap out of some list of $n$ elements, and the removing a maximal value one at a time.

The Algorithm

The following pseudocode illustrates the heapsort algorithm. It builds upon the heap insertion and heap removal algorithms.

Algorithm HeapSort($\mathrm{(}\mathrm{A}\mathrm{,}\mathrm{⪯}\mathrm{,}\mathrm{n}\mathrm{)}$)
Input: List $A$ of $n$ elements
Output: $A$ sorted, such that $⪯$ is a total order over $A$

begin
         for $i\leftarrow 2\mathrm{𝐭𝐨𝐧}$ do

HeapInsert$\mathrm{(}A\mathrm{,}\mathrm{⪯}\mathrm{,}i\mathrm{-}\mathrm{1}\mathrm{,}A\mathbf{}\mathrm{[}i\mathrm{]}\mathrm{)}$

for $i\leftarrow n\mathrm{𝐝𝐨𝐰𝐧𝐭𝐨𝟐}$ do

$A[i-1]\leftarrow \mathrm{𝐇𝐞𝐚𝐩𝐑𝐞𝐦𝐨𝐯𝐞}(𝐇,𝐢,⪯)$

end

Analysis

Note that the algorithm given is based on a top-down heap insertion algorithm.It is possible to get better results through bottom-up heap construction.

Each step of each of the two for loops in this algorithm has a runtime complexity of $𝒪(\log i)$. Thus overall the heapsort algorithm is $𝒪(n\log n)$.

Heapsort is not quite as fast as quicksort in general, but it is not much slower, either. Also, like quicksort, heapsort is an in-place sorting algorithm, but not a stable sorting algorithm.Unlike quicksort, its performance is guaranteed, so despite the ordering of its input its worst-case complexity is $𝒪(n\log n)$.Given its simple implementation and reasonable performance, heapsort is ideal for quickly implementing a decent sorting algorithm.