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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)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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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).
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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