Count inversions in an array efficiently
Problem Description
Given an array of integers, compute the total number of "inversion pairs". An inversion pair (i, j) is defined as a pair of indices such that i < j and the element at index i is greater than the element at index j (arr[i] > arr[j]).
For example, given the array [5, 7, 9, 2, 3, 12, 8, 4], the inversion pairs include (5,2), (5,3), (5,4), (7,2), (7,3), (7,4), (9,2), (9,3), (9,4), (9,8), (12,8), (12,4), (8,4).
Requirements
- Naive Approach: Describe an O(n^2) algorithm to count inversion pairs.
- Efficient Approach: Describe an O(n log n) divide-and-conquer algorithm to count inversion pairs.
- Complexity Analysis: For both approaches, analyze their time and space complexities.
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Core dilemma
Optimizing a brute-force quadratic solution to a more efficient logarithmic one by leveraging divide-and-conquer principles, specifically how to count inversions accurately during the merge step of a sorting algorithm.
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Count inversions in an array efficiently
Problem Description
Given an array of integers, compute the total number of "inversion pairs". An inversion pair (i, j) is defined as a pair of indices such that i < j and the element at index i is greater than the element at index j (arr[i] > arr[j]).
For example, given the array [5, 7, 9, 2, 3, 12, 8, 4], the inversion pairs include (5,2), (5,3), (5,4), (7,2), (7,3), (7,4), (9,2), (9,3), (9,4), (9,8), (12,8), (12,4), (8,4).
Requirements
- Naive Approach: Describe an O(n^2) algorithm to count inversion pairs.
- Efficient Approach: Describe an O(n log n) divide-and-conquer algorithm to count inversion pairs.
- Complexity Analysis: For both approaches, analyze their time and space complexities.