By Herbert S. Wilf
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Additional resources for Algorithms and Complexity (Internet edition, 1994)
1 illustrates the claim. Fig. 1: Conditions (a), (b), (c) To see this, observe first that (a), (b), (c) are surely true at the beginning, when j = left + 1. Next, if for some j they are true, then the execution of lines 7, 8 guarantee that they will be true for the next value of j. Now look at (a), (b), (c) when j = right. It tells us that just prior to the execution of line 9 the condition of the array will be (a) x[left] = T and (b) x[r] < T for all left < r ≤ i and (c) x[r] ≥ T for all i < r ≤ right.
Possible orders in which these elements might appear, so we are considering the action of Quicksort on just these n! inputs. Second, for each particular one of these inputs, the choices of the splitting elements will be made by choosing, at random, one of the entries of the array at each step of the recursion. We will also average over all such random choices of the splitting elements. Therefore, when we speak of the function F (n), the average complexity of Quicksort, we are speaking of the average number of pairwise comparisons of array entries that are made by Quicksort, where the averaging 35 Chapter 2: Recursive Algorithms is done first of all over all n!
In other words, if we can reduce the number of multiplications of numbers that are needed to multiply two 2 × 2 matrices, then that improvement will show up in the exponent of N when we measure the complexity of multiplying two N × N matrices.