Introduction

In this blog post, we will explore how to solve the LeetCode problem "775" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

Problem Statement

You are given an integer array nums of length n which represents a permutation of all the integers in the range [0, n - 1]. The number of global inversions is the number of the different pairs (i, j) where: 0 <= i < j < n nums[i] > nums[j] The number of local inversions is the number of indices i where: 0 <= i < n - 1 nums[i] > nums[i + 1] Return true if the number of global inversions is equal to the number of local inversions. Example 1: Input: nums = [1,0,2] Output: true Explanation: There is 1 global inversion and 1 local inversion. Example 2: Input: nums = [1,2,0] Output: false Explanation: There are 2 global inversions and 1 local inversion. Constraints: n == nums.length 1 <= n <= 105 0 <= nums[i] < n All the integers of nums are unique. nums is a permutation of all the numbers in the range [0, n - 1].

Explanation

Here's the breakdown:

• Key Idea: The problem states that nums is a permutation of [0, n-1]. If the number of global and local inversions are equal, it means that any global inversion must also be a local inversion. In other words, if nums[i] > nums[j] where i < j, then j must be equal to i + 1. This implies that if the global and local inversions are equal, the array must be "almost sorted" meaning that it can be brought into sorted order using only adjacent swaps. A global inversion that is not a local inversion would involve elements that are not adjacent. Thus we must detect if such a situation exists to determine if the number of local and global inversions can be equal.
• Simplification: A non-local inversion (global inversion where elements are not adjacent) exists if there is an index i such that nums[i] > nums[j] where j > i + 1. This means nums[i] must be greater than at least one element that is located beyond the immediately next element to it. We can simply iterate through the array and check for such violation to determine if global inversions can possibly equal local inversions.

• Early Exit: If at any point we find such an i that violates the 'almost sorted' condition, we can immediately return False. Otherwise, if we complete the loop, it means that the array satisfies the 'almost sorted' property and we can return True.

• Complexity: O(n) runtime, O(1) storage.

Code

    def isIdealPermutation(nums):
    """
    Checks if the number of global inversions is equal to the number of local inversions.

    Args:
        nums: An integer array of length n representing a permutation of all the integers in the range [0, n - 1].

    Returns:
        True if the number of global inversions is equal to the number of local inversions, False otherwise.
    """
    n = len(nums)
    for i in range(n - 2):
        if nums[i] > nums[i + 2]:
            return False
    return True