The documentation for heapq.heapify indicates that it runs in linear time. I believe that this is incorrect, and that it runs in worst case n * log(n) time. I checked the implementation, and there are indeed n _siftup operations, which each appear to be worst case log(n).

One example of the documentation pages that are wrong.
http://docs.python.org/3.4/library/heapq.html#heapq.heappush