为何heapify时间复杂度为O(n)，heappush为O(log n)？heapify等价n次heappush吗？

Great question—this is a super common gotcha when working with heaps, especially in Python's heapq module. Let's unpack this clearly.

1. The Time Complexity Breakdown #

Heappush is O(log n) #

When you call heapq.heappush(arr, element), here's what happens:

• The element gets added to the end of the list (O(1) operation).
• Then, it has to bubble up (officially called sift up) to its correct position in the heap. This means comparing it with its parent node and swapping if needed, repeating until it's in a spot where the heap property holds.

Since a heap is a complete binary tree, the height of a heap with n elements is log2(n). The bubble-up process can take at most this many steps—hence the O(log n) time complexity per push.

Heapify is O(n) #

heapq.heapify(arr) works completely differently. Instead of building the heap from the ground up by adding elements one by one, it starts from the last non-leaf node and works its way up to the root, performing a sift down operation on each node.

Here's why this is linear time:

• Most nodes in a complete binary tree are leaves (about half of them), and leaves don't need any sifting—they're already valid heap nodes.
• Nodes in the second-to-last level only need to sift down 1 level at most.
• Nodes in the third-to-last level need at most 2 levels, and so on, until the root, which might need to sift down log2(n) levels.

When you sum up all these potential operations, the total number of comparisons/swaps ends up being O(n). The mathematical proof shows this sum converges to less than 2n, so it's a linear-time operation, not O(n log n).

2. Is heapify equivalent to n heappush calls? #

No, absolutely not—and here's why:

• Time efficiency: If you started with an empty list and called heappush n times to add all elements, that would take O(n log n) time total, which is way slower than heapify's O(n).
• Resulting heap structure: Even if you end up with a valid heap in both cases, the actual arrangement of elements in the list can be different. For example:
  • Take the list [3, 2, 1]. Running heapq.heapify(arr) gives you [1, 2, 3].
  • If you start with an empty list and push each element in order:
    1. Push 3 → [3]
    2. Push 2 → [2, 3]
    3. Push 1 → [1, 3, 2]
       The end result is a valid heap, but the element positions are different from the heapify output.

Heapify is a highly optimized operation designed to turn an existing list into a heap in linear time, which is far more efficient than building the heap incrementally with repeated heappush calls.

内容的提问来源于stack exchange，提问作者Maximus S