Lecture No.14
4.3.5 Average-case Analysis of Quicksort
We will now show that in the average case, quicksort runs in Θ(n log n) time. Recall that
when we talked about average case at the beginning of the semester, we said that it
depends on some assumption about the distribution of inputs. However, in the case of
quicksort, the analysis does not depend on the distribution of input at all. It only depends
upon the random choices of pivots that the algorithm makes. This is good, because it
means that the analysis of the algorithm’s performance is the same for all inputs.
In this case the average is computed over all possible random choices that the algorithm
might make for the choice of the pivot index in the second step of the QuickSort
procedure above.
To analyze the average running time, we let T(n) denote the average running time of
QuickSort on a list of size n. It will simplify the analysis to assume that all of the
elements are distinct. The algorithm has n random choices for the pivot element, and each
choice has an equal probability of 1/n of occurring. So we can modify the above
recurrence to compute an average rather than a max, giving:
The time T(n) is the weighted sum of the times taken for various choices of q. I.e.,
T(n) = [ 1/n ( T(0) + T(n - 1) + n ) + 1/n ( T(1) + T(n - 2) + n )
+ 1/n ( T(2) + T(n - 3 ) + n ) + · · · + 1/n (T(n - 1) + T(0) + n)I ]
We have not seen such a recurrence before. To solve it, expansion is possible but it is
rather tricky. We will attempt a constructive induction to solve it. We know that we want
a _(n log n). Let us assume that T(n) ≤ cn log n) for n ≥ 2 where c is a constant.
For the base case n = 2 we have
We want this to be at most c2 log 2, i.e.,
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