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The heap sort algorithm is as follows:

1. Call the build_max_heap() function.

2. Swap the first and last elements of the max heap.

3. Reduce the heap by one element (elements that follow the heap are in sorted order).

4. Call the sift_down() function.

5. Goto step 2 unless the heap has one element.

The build_max_heap() function creates the max heap and takes linear time, O(n). The sift_down() function moves the first element in the heap into its correct index, thus restoring the max heap property. This takes O(log(n)) and is called n times, so takes O(n * log(n)). The complete algorithm therefore equates to O(n + n * log(n)).

If you start with a max heap rather than an unsorted array, there will be no difference in the runtime because the build_max_heap() function will still take O(n) time to complete. However, the mere fact you are starting with a max heap means you must have built that heap prior to calling the heap sort algorithm, so you've actually increased the overall runtime by an extra O(n), thus taking O(2n * log(n)) in total.

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