4. Core Operations and the Heap Sort Algorithm

The Heap Sort algorithm is a two-phase process that masterfully uses the properties of the Max-Heap. Both phases rely on a core "helper" operation that maintains the heap property.

The Engine of the Heap: The siftdown Operation

To build and maintain a heap, we need a procedure to fix any violations of the heap property. The primary operation used in Heap Sort is siftdown (also known as heapify-down).

• siftdown (or Heapify-Down): This operation is used when a node's value is smaller than one of its children, violating the Max-Heap property. The siftdown process corrects this by repeatedly swapping the parent with its largest child, effectively "sinking" the smaller element down the tree until the heap property is restored for that subtree. The time complexity for a single siftdown operation is proportional to the height of the tree, making it O(log n).

• siftup (or Heapify-Up): While less critical for our Heap Sort implementation, this complementary operation is used when a newly inserted node is larger than its parent. It "bubbles" the element up the tree by swapping it with its parent until the heap property is restored. This is the key operation used when adding elements to a std::priority_queue.

With the siftdown operation as our main tool, we can now construct the two phases of Heap Sort.

Phase 1: makeheap (The Heapify Process)

The first step is to convert the unsorted input array into a valid Max-Heap.

• Goal: Rearrange the elements of the array so that they satisfy the Max-Heap property.

• Method (Bottom-Up): The most efficient way to do this is with a bottom-up approach. We treat the entire array as a complete binary tree and then iteratively fix it. The process starts from the last non-leaf node and moves upwards towards the root. For each node, we call siftdown to ensure its subtree is a valid Max-Heap. By the time we reach the root, the entire array is guaranteed to be a Max-Heap.

• Analysis: While it involves multiple calls to siftdown (an O(log n) operation), a tight analysis shows that the makeheap phase can be completed in O(n) linear time, which is remarkably efficient.

Phase 2: The Sorting Process

Once the array is a Max-Heap, the largest element is at the root (array[0]). The sorting phase systematically extracts this largest element and places it in its correct final position.

This is done through a repeated process:

1. Swap: Swap the root element (array[0], the current maximum) with the last element in the heap portion of the array. The largest element is now in its final, sorted position at the end of the array.

2. Shrink: The effective size of the heap is reduced by one, "locking in" the sorted element at the end so it is no longer considered part of the heap.

3. Repair: The new root element (which was previously the last element) likely violates the Max-Heap property. Call siftdown on the root (array[0]) to repair the heap, ensuring the next largest element rises to the top.

This cycle is repeated n-1 times, until the entire array is sorted.

Complexity Analysis of Heap Sort

• Time Complexity:

  • Phase 1 (makeheap): O(n)

  • Phase 2 (Sorting): Consists of n-1 calls to siftdown, each taking O(log n) time. Total time for this phase is O(n log n).

  • The overall complexity is dominated by the sorting phase, giving Heap Sort a guaranteed Θ(n log n) time complexity for worst-case, average-case, and best-case scenarios.

• Space Complexity:

  • The algorithm operates directly on the input array, swapping elements within it. It requires no significant extra storage. Therefore, Heap Sort is an in-place algorithm with a space complexity of O(1).