# 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($(A,\preceq,n)$)
Input: List $A$ of $n$ elements
Output: $A$ sorted, such that $\preceq$ is a total order over $A$

begin
for $i\leftarrow 2\bf{to}n$ do

HeapInsert$(A,\preceq,i-1,A[i])$

for $i\leftarrow n\bf{downto}2$ do

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

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 $\mathcal{O}(\log i)$. Thus overall the heapsort algorithm is $\mathcal{O}(n\log n)$.

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 $\mathcal{O}(n\log n)$. Given its simple implementation and reasonable performance, heapsort is ideal for quickly implementing a decent sorting algorithm.

Title heapsort Heapsort 2013-03-22 12:31:02 2013-03-22 12:31:02 mathcam (2727) mathcam (2727) 9 mathcam (2727) Algorithm msc 68P10 HeapInsertionAlgorithm HeapRemovalAlgorithm Heap SortingProblem