Why is heapify linear?
Josiah Carlson
jcarlson at uci.edu
Mon Nov 1 23:38:25 CET 2004
Scott David Daniels <Scott.Daniels at Acm.Org> wrote:
>
> I am sorry, but in the Python 2.4 description of "heapify", I find the
> description of "Transform list x into a heap, in-place, in linear time,"
> unbelievable. I understand the hand-wave that makes dictionary building
> linear (though I have a hard time with even that). Could somebody tell
> me what twist of logic makes "heapify" linear, except in the sense that
> linear is coming to mean "very fast?"
It has to do with the amount of work done by doing the heapify operation
from the bottom up.
Using ugly ascii-art math, we get the following work for heapifying from
the bottom up:
log(n) i*n
SUM ----
i=1 2**i
That sum is known to converge to 2n from below (there is a simple
picture that I can present that will prove it to you, if desired). 2n is
O(n). Hence linear.
- Josiah
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