Reverse Pairs

Problem Description

Given an integer array nums, return the number of reverse pairs in the array. A reverse pair is a pair (i, j) where 0 <= i < j < nums.length and nums[i] > 2 * nums[j].

Key Insights

• A brute force approach would involve checking each pair of indices, leading to O(n^2) time complexity.
• We can utilize a divide-and-conquer strategy, similar to merge sort, to count reverse pairs efficiently.
• By counting the number of valid pairs during the merge step, we can achieve O(n log n) time complexity.
• The problem can also be addressed using data structures like Binary Indexed Trees or Segment Trees for dynamic counting.

Space and Time Complexity

Time Complexity: O(n log n)
Space Complexity: O(n)

Solution

To solve the problem efficiently, we can leverage a modified merge sort algorithm. The steps are as follows:

1. Divide the array into two halves.
2. Count reverse pairs in each half recursively.
3. Count the number of reverse pairs that span both halves while merging them back together.
4. During the merge step, for each element in the left half, we will find how many elements from the right half satisfy the condition nums[i] > 2 * nums[j].
5. This counting can be done efficiently as both halves are sorted.

Code Solutions