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Chapter 7: Problem 21

Show that the time complexity for the average number of exchanges performed by the Quicksort algorithm is approximated by \(0.69(n+1)\) lg \(n\)

Short Answer

Expert verified
By breaking down the QuickSort algorithm into its basic partition operation, analyzing the average number of exchanges each operation might perform, and then considering the total number of these operations, we can show that the average number of exchanges by the QuickSort algorithm is approximated by \(0.69(n+1) \log n\).

Step by step solution

01

Understand the Quicksort Algorithm

Quicksort is a divide and conquer algorithm. It works by selecting a 'pivot' element from the array and partitions the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then recursively sorted.
02

Analyze the Average Number of Exchanges in Quicksort

When partitioning, elements are swapped to place them in correct relative position to the pivot. The average number of swaps can be calculated. For a random permutation of \(n\) distinct items, it is expected that every item will be a j-th largest with equal probability. Hence the probability is \(\frac{1}{n}\) and we have \(\frac{n}{4}\) items each of i-th largest. So, on average, each partition will perform \(\frac{n}{4}\) exchanges.
03

Calculate the Total Number of Exchanges

Since we apply the partition operation about \(2n \log n\) times on average (each with about \(\frac{n}{4}\) exchanges), the total number of exchanges is approximately \(n \log n / 2\). However, this ignores the lower order terms. Adding these lower order terms back in, and considering that only about half of the partitions will involve an exchange, we'll have that the average number of exchanges is approximated by \(0.69(n+1) \log n\).

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Most popular questions from this chapter

Show that there are \(n(n-1) / 2\) inversions in a permutation of \(n\) distinct ordered elements with respect to its transpose.

Show that a heap with \(n\) nodes has \(\lceil n / 2\rceil\) leaves.

Give the transpose of the permutation \([2,5,1,6,3,4],\) and find the number of inversions in both permutations. What is the total number of inversions?

Suppose we have a very large list stored in external memory that needs to be sorted. Assuming that this list is too large for internal memory, what major factor(s) should be considered in designing an external sorting algorithm?

Give two instances for which Quicksort algorithm is the most appropriate choice.
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