The efficiency of quicksort is determined by calculating the running time
of the two recursive calls plus the time spent in the partition. The
partition step of quicksort takes *n* - 1 comparisons. The efficiency of the
recursive calls depends largely on how equally the pivot value splits the
array. In the average case, assume that the pivot does split the array
into two roughly equal halves. As is common with divide-and-conquer sorts,
the dividing algorithm has a running time of *log*(*n*). Thus the overall
quicksort algorithm has running time *O*(*nlog*(*n*)). The worst case occurs
when the pivot value always ends up being one of the extreme values in the
array. For example, this might happen in a sorted array if the first value
is selected as the pivot. In this case, the partitioning phase still
requires n-1 comparisons, as before, but quicksort does not achieve the
*O*(*log*(*n*)) efficiency in the dividing process. Instead of breaking an 8
element array into arrays of size 4, 2, and 1 in three recursive calls, the
array size only reduces by one: 7, 6, and 5. Thus the dividing process
becomes linear and the worst case efficiency is *O*(*n*^{2}). Note that quicksort
performs badly once the amounts of data become small due to the overhead of
recursion. This is often addressed by switching to a different sort for
data smaller than some magic number such as 25 or 30 elements.

Take a Study Break!