Understanding the Time Complexity of Heapsort

Preview

1. Building the Heap:
   • The process of building a heap from an array of $n$ elements can be done in $O(n)$ time. This is achieved by starting from the middle of the array and working backwards, performing a "heapify" operation on each element to ensure it satisfies the heap property.
2. Sorting the Heap:
   • Once the heap is built, the sorting phase involves repeatedly removing the maximum element (root of the heap) and placing it at the end of the array, then restoring the heap property. Each removal and heapify operation takes $O(\log n)$ time. Since this is done $n$ times, the total time complexity for this phase is $O(n \log n)$.

Detailed Explanation

• Heapify Operation: The heapify operation ensures that a node and its children satisfy the heap property. This operation takes $O(\log n)$ time because it may need to traverse from the node to the root of the heap.
• Building the Heap: Starting from the middle of the array and moving backwards, each node is heapified. The total time for this process is $O(n)$.
  

  

  Preview
• Sorting Phase: Each removal of the root (maximum element) and subsequent heapify operation takes $O(\log n)$ time. Since this is done $n$ times, the total time for this phase is $O(n \log n)$.