Final answer:
The question asks for an algorithm to count inversions in a permutation of n elements efficiently. The modified merge sort algorithm does this in θ(n log n) time by dividing and conquering, recursively counting the inversions in each half, and during the merge step.
Step-by-step explanation:
The student is asking for an algorithm to determine the number of inversions in a permutation of n elements that runs in θ(n log n) worst-case time. An inversion is a pair of indices i, j such that i < j and A[i] > A[j], where A is an array representing the permutation.
One commonly used algorithm for this purpose is a modification of the merge sort algorithm, which sorts the array and counts the inversions in the process.
The algorithm works by dividing the array into two halves, recursively counting the inversions within each half, and then counting inversions that occur across the two halves during the merge step.
Each step takes linear time in the size of the subarrays being merged, and since there are log(n) levels of recursion in a merge sort, the overall complexity is θ(n log n).
The detailed step-by-step algorithm is:
Divide the array into two halves.
Recursively count inversions in the left half.
Recursively count inversions in the right half.
Merge the two halves together while counting cross inversions.
Add the number of inversions from the left and right halves and the number of cross inversions.
Return the total number of inversions.
This merge sort-based algorithm efficiently computes the number of inversions, providing valuable information on the 'sortedness' of an array.