Problem Description

In this problem, you are provided with an integer array called nums, where the array is indexed starting from 0. You must identify all pairs of indices (i, j) that qualify as a "bad pair". A pair is designated as a bad pair if it satisfies two conditions:

1. i < j (the index i comes before index j)
2. j - i is not equal to nums[j] - nums[i]

This means that the relative difference between the indices should equal the relative difference between their respective elements for it to be a good pair; otherwise, it is considered bad.

Your task is to calculate and return the total count of bad pairs in the nums array.

Intuition

The brute-force approach would be to check all possible pairs (i, j) where i < j and count the number of pairs where j - i does not equal nums[j] - nums[i]. However, this approach has a time complexity of O(n^2) which would not be efficient for large arrays.

To optimize this, we can make a key observation: instead of focusing on whether a pair is bad, concentrate on finding the good pairs because they adhere to a specific pattern, namely j - i = nums[j] - nums[i]. This can be rearranged to j - nums[j] = i - nums[i]. If this property holds true for any two indices, they form a good pair. If it doesn't, they form a bad pair.

The intuition for the solution is to transform the problem such that we keep track of how many times we have seen the same difference (i - nums[i]) before. When we process the array, at each index i, we calculate this difference and check it against a count tracker. For a new difference value at index i, all previous occurrences of this value (the count tracked so far) represent the number of good pairs with the new index i. Hence, bad pairs can be determined by subtracting the number of good pairs at each step from the total number of pairs seen so far.

The Counter dictionary in the solution keeps track of how many times we have seen each difference. For each new element x at index i, we increment the counter for i - x which indicates we've seen another instance where this difference occurs. We then update the answer by adding i (the total number of pairs with the current index so far) minus the count of this difference (the number of good pairs). The final answer reflects the total number of bad pairs.

This solution provides an efficient way to calculate the number of bad pairs with a time complexity of O(n), which is a significant improvement over the brute-force method.

Solution Approach

The implementation uses a hash map to track the counts of the differences. In Python, this is conveniently handled using the Counter class from the collections module. The Counter works like a standard dictionary but is specialized for counting objects. It has methods that make it easy to increment counts for each object.

The key step in the algorithm is to iterate over the array nums while using the enumerate function that provides both the index i and the value x. The difference i - x is the transformed value we are interested in. Here's how the algorithm proceeds:

1. Initialize a Counter object cnt that will track the counts of differences.

2. Initialize ans to 0, which will keep track of the number of bad pairs.

3. Loop through the nums array using an enumerate loop to get both the index i and the value x at that index.

4. For each element, the difference i - x is calculated, which will be used to determine if a pair is good.

   • ans is incremented by i - cnt[i - x], which represents the number of bad pairs up to the current index i. This is calculated by subtracting the number of good pairs (tracked by cnt[i - x]) from the total number of pairs that could have been formed with all the previous elements (given by i).

5. The count for the current difference i - x is incremented in cnt to record that we have seen another instance where this difference occurs.

6. After the loop ends, ans contains the total count of bad pairs, which is returned.

The reason why this approach works is due to the transformation of the problem into a more manageable counting problem. By tracking the occurrence of the differences, we avoid directly comparing every possible pair, which results in a much more efficient algorithm.

The key components of the implementation are:

• Enumeration: Providing both the index and the value to easily calculate our necessary difference.
• Hash Map (Counter): For efficient storage and retrieval of counts associated with each difference. This allows us to efficiently keep track of the number of good pairs per difference.
• Transformation: Condensing the condition from j - i != nums[j] - nums[i] to i - nums[i], making it easier to track and count through a hash map.

By utilizing these components, we achieve a linear time solution, which is optimal for this problem.

Example Walkthrough

Let's illustrate the solution approach using a small example. Consider the nums array as [1, 2, 3, 5]. We want to find all bad pairs where i < j and j - i is not equal to nums[j] - nums[i].

First, we initialize a Counter object cnt and a variable ans to count the bad pairs, starting with ans set to 0.

Now, we start iterating through the array with indices and values:

• At i = 0, x = 1, the difference i - x is -1. This difference hasn't been seen before, so cnt[-1] is 0. We update ans with 0 - 0 = 0, and increment cnt[-1].
• At i = 1, x = 2, the difference i - x is -1 again. Now, cnt[-1] is 1 (from the previous count). We update ans with 1 - 1 = 0, but we already have 0 bad pairs. Thus, ans stays as 0, and we increment cnt[-1] to 2.
• At i = 2, x = 3, the difference i - x is -1 yet again. cnt[-1] is now 2. We update ans with 2 - 2 = 0, and the number of bad pairs ans stays at 0. We increment cnt[-1] to 3.
• At i = 3, x = 5, the difference i - x is -2. This is a new difference, so cnt[-2] is 0. We update ans with 3 - 0 = 3, indicating that up to this index, we have 3 bad pairs. We increment cnt[-2].

By the end of the loop, we have counted ans = 3 bad pairs. To confirm, we can check the conditions for each pair:

• (0, 1): 1 - 0 = 2 - 1 (Good pair)
• (0, 2): 2 - 0 = 3 - 1 (Good pair)
• (0, 3): 3 - 0 ≠ 5 - 1 (Bad pair)
• (1, 2): 2 - 1 = 3 - 2 (Good pair)
• (1, 3): 3 - 1 ≠ 5 - 2 (Bad pair)
• (2, 3): 3 - 2 ≠ 5 - 3 (Bad pair)

There are indeed 3 bad pairs: (0, 3), (1, 3), (2, 3).

The counter at the end of processing will be { -1: 3, -2: 1 }, reflecting the idea that negative counts corresponded to instances where a good pair did not occur, leading to bad pairs.

This example shows how the algorithm can effectively identify bad pairs by focusing on counting the occurrences of the i - nums[i] and using that to quickly determine the number of bad pairs without comparing each pair directly.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(n), where n is the length of the nums array. The reason for this is that the code uses a single loop that iterates through all elements of nums, performing a constant amount of work within each iteration. The cnt Counter dictionary lookup and update operations are O(1) on average, so they don't change the overall linear time complexity.

The space complexity of the code is also O(n) because the cnt Counter dictionary can potentially grow to have as many distinct keys as there are elements in nums. In the worst case, each iteration introduces a new key-value pair into the cnt dictionary.

Learn more about how to find time and space complexity quickly using problem constraints.