Steven D'Aprano wrote:
Big Oh notation is good for estimating asymptotic behaviour, which means it
is good for predicting how an algorithm will scale as the size of the input
increases. It is useless for predicting how fast that algorithm will run,
since the actual speed depends on those constant factors that Big Oh
neglects. That's not a criticism of Big Oh -- predicting execution speed is
not what it is for.
For what it's worth, for very small lists, the while...remove algorithm is
actually faster that using a list comprehension, despite being O(M*N)
versus O(N), at least according to my tests. It's *trivially* faster, but
if you're interested in (premature) micro-optimisation, you might save one
or two microseconds by using a while loop for short lists (say, around
N<=12 or so according to my tests), and swapping to a list comp only for
larger input.
Now, this sounds silly, and in fact it is silly for the specific problem
we're discussing, but as a general technique it is very powerful. For
instance, until Python 2.3, list.sort() used a low-overhead but O(N**2)
insertion sort for small lists, only kicking off a high-overhead but O(N)
O(NlogN)
sample sort above a certain size. The current timsort does something
similar, only faster and more complicated. If I understand correctly, it
uses insertion sort to make up short runs of increasing or decreasing
values, and then efficiently merges them.
It uses binary insertion sort to make runs of 64 (determined by
empirical testing). The 'binary' part means that it uses O(logN) binary
search rather than O(n) linear search to find the insertion point for
each of N items, so that finding insertion points uses only O(NlogN)
comparisions (which are relatively expensive). Each of the N insertions
is done with a single memmove() call, which typically is relatively
fast. So although binary insertion sort is still O(N*N), the
coefficient of the N*N term in the time formula is relatively small.
The other advantages of timsort are that it exploits existing order in a
list while preserving the order of items that compare equal.
Terry Jan Reedy
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