[Python-Dev] On time complexity of heapq.heapify

[Rafael Almeida]

Hello,

Current heapify documentation says it takes linear time

    https://docs.python.org/3/library/heapq.html#heapq.heapify

However, investigating the code (Python 3.5.2) I saw this:

    def heapify(x):
        """Transform list into a heap, in-place, in O(len(x)) time."""
        n = len(x)
        # Transform bottom-up.  The largest index there's any point to
looking at
        # is the largest with a child index in-range, so must have 2*i + 1
< n,
        # or i < (n-1)/2.  If n is even = 2*j, this is (2*j-1)/2 = j-1/2 so
        # j-1 is the largest, which is n//2 - 1.  If n is odd = 2*j+1, this
is
        # (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1.
        for i in reversed(range(n//2)):
            _siftup(x, i)

>From what I gather, _siftup(heap, pos) does not run in constant time, but
rather it runs in time proportional to the height of the subtree with root
in ``pos''. Although, according to the in-code comments, it should be
faster than creating a heap by calling heappush multiple times, I believe
the time complexity remains the same.

Although I had a hard time finding out the exact time complexity for that
particular function, I think it is closer to O(log(n!)) than to O(n). I
would be very happy to see an explanation as to why the time is O(n) (it
does not seem possible to me to create a heap of n numbers in such
runtime). However, if no one has a convincing argument, I'd rather omit
time complexity information from documentation (given that this analysis is
not made for the other functions either).

[]'s
Rafael
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