Let A[1..n] be an array of n distinct numbers. If i A[j], then the pair (i, j) is called an inversion of A. This problem appears in the real-life scenario below. Music (or movie) sites try to match your song preferences with others: you rank n songs, music site consults database to find people with similar tastes. Similarity metric: number of inversions between two rankings. My rank: 1, 2, ..., n; Your rank: a₁, A2, ..., an; Songs i and j are inverted if i aj. a. List the five inversions of the array (2, 3, 8, 6, 1) b. What array with elements from the set {1, 2, ..., n} has the most inversions? How many does it have?
Let A[1..n] be an array of n distinct numbers. If i A[j], then the pair (i, j) is called an inversion of A. This problem appears in the real-life scenario below. Music (or movie) sites try to match your song preferences with others: you rank n songs, music site consults database to find people with similar tastes. Similarity metric: number of inversions between two rankings. My rank: 1, 2, ..., n; Your rank: a₁, A2, ..., an; Songs i and j are inverted if i aj. a. List the five inversions of the array (2, 3, 8, 6, 1) b. What array with elements from the set {1, 2, ..., n} has the most inversions? How many does it have?
Computer Networking: A Top-Down Approach (7th Edition)
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![Let A[1..n] be an array of n distinct numbers. If i <j and A[i] > A[j], then the pair (i, j) is called
an inversion of A.
This problem appears in the real-life scenario below.
Music (or movie) sites try to match your song preferences with others: you rank n songs,
music site consults database to find people with similar tastes.
Similarity metric: number of inversions between two rankings. My rank: 1, 2,
rank: a₁, a₂, ..., an; Songs i and j are inverted if i <j, but a¡ > aj.
***/
a. List the five inversions of the array (2, 3, 8, 6, 1)
b. What array with elements from the set {1, 2, . n} has the most inversions? How many
does it have?
c. A Brute force solution to this problem is to check all (n²) pairs i and j.
Give a divide-and-conquer algorithm that determines the number of inversions in any
permutation on n elements in O(nlogn) worst-case time. (Hint: Modify MergeSort).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9be7f291-8575-4463-9f30-32219116b9fd%2F9b7c1783-1322-4b46-a214-198a40b4816d%2F3svg88_processed.png&w=3840&q=75)
Transcribed Image Text:Let A[1..n] be an array of n distinct numbers. If i <j and A[i] > A[j], then the pair (i, j) is called
an inversion of A.
This problem appears in the real-life scenario below.
Music (or movie) sites try to match your song preferences with others: you rank n songs,
music site consults database to find people with similar tastes.
Similarity metric: number of inversions between two rankings. My rank: 1, 2,
rank: a₁, a₂, ..., an; Songs i and j are inverted if i <j, but a¡ > aj.
***/
a. List the five inversions of the array (2, 3, 8, 6, 1)
b. What array with elements from the set {1, 2, . n} has the most inversions? How many
does it have?
c. A Brute force solution to this problem is to check all (n²) pairs i and j.
Give a divide-and-conquer algorithm that determines the number of inversions in any
permutation on n elements in O(nlogn) worst-case time. (Hint: Modify MergeSort).
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