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Chapter 7: Problem 10

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?

Short Answer

Expert verified
The transpose of the permutation is \([2,5,1,6,3,4]\). The number of inversions in both permutations is 6 and the total inversions is 12.

Step by step solution

01

Determining the Transpose

The transpose of a permutation is found by interchanging the rows and columns. Given the permutation \([2,5,1,6,3,4]\), the first element becomes the first element of the transposed permutation, the second element becomes the second and so on. This gives the transposed permutation as \([2,5,1,6,3,4]\).
02

Calculating Inversions for Permutations

An inversion in a permutation occurs when a larger number precedes a smaller one. For the given permutation \([2,5,1,6,3,4]\), inversions are: (2,1), (5,1), (5,3), (5,4), (6,3), (6,4) which makes it 6 inversions. Similarly, for the transpose \([2,5,1,6,3,4]\), inversions are same and also 6.
03

Summing Up the Inversions

The total number of inversions is the sum of the inversions in the original permutation and its transpose. So the total inversions is \(6 + 6 = 12\).

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Most popular questions from this chapter

Write an algorithm that sorts a list of \(n\) elements in nonincreasing order by finding the largest and smallest elements and exchanges those elements with the elements in the first and last positions. Then the size of the list is reduced by \(2,\) excluding the two elements that are already in the proper positions, and the process is repeated on the remaining part of the list until the entire list is sorted. Analyze your algorithm and show the results using order notation.

In the process of rebuilding the master list, the Radix Sort algorithm (Algorithm 7.6) wastes a lot of time examining empty sublists when the number of piles (radix) is large. Is it possible to check only the sublists that are not empty?

Show that the permutation \([n, n-1, \ldots, 2,1]\) has \(n(n-1)\) 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?

Show that the worst-case and average-case time complexities for the number of assignments of records performed by the Insertion Sort algorithm (Algorithm 7.1) are given by $$W(n)=\frac{(n+4)(n-1)}{2} \approx \frac{n^{2}}{2} \quad \text { and } \quad A(n)=\frac{n(n+7)}{4}-1 \approx \frac{n^{2}}{4}$$
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