Problem Description

The problem presents us with an integer array called nums, indexed from 0, which means the first element of the array is at index 0. Our task is to calculate a new array, diff, also of the same length as nums, based on a specific rule. For each position i in the nums array, we have to find the count of distinct elements in two different parts of nums:

1. The "prefix" which includes all elements from the start of nums up to and including the element at index i.
2. The "suffix" which includes all elements after index i up to the end of the nums.

diff[i] is then defined as the number of distinct elements in the prefix minus the number of distinct elements in the suffix.

Note that when working with subarrays or slices of an array, nums[i, ..., j] means we are considering the elements of nums from index i through to index j, inclusive. If i is greater than j, it means the subarray is empty.

Intuition

To construct the diff array as described, we need to keep track of the distinct elements as we move through the nums array. Using two sets, we can capture the unique elements in the prefixes and suffixes of nums at each index.

The algorithm includes the following steps:

1. Initialize an extra array suf with the same length as nums plus one, to record the number of distinct elements in the suffix part starting from each index. We add an extra space since we are including the empty subarray when i > j.

2. Traverse the nums array in reverse (right to left), starting from the last element, and with each element, we encounter, add it to a set to ensure only distinct elements are counted. Update the corresponding suf entry with the length of this set to reflect the current number of distinct elements in the suffix up to the current index.

3. Clear the set (for counting distinct elements in the prefixes) and then traverse the nums array once more, from left to right. With each element, we add it to the set to keep track of distinct elements and calculate diff[i] by subtracting the number of distinct elements in the suffix (obtained from suf[i + 1]) from the number of distinct elements in the prefix (length of the set at this point in the traversal).

This approach allows us to incrementally build up the distinct element counts for the prefix and suffix arrays, which can then be used to quickly compute the diff array without having to re-check all the elements at each index.

Solution Approach

The solution uses a couple of Python data structures - lists and sets - and leverages their properties to solve the problem efficiently:

1. Lists: They are used to store the input (nums) and output (ans for the distinct difference array, and suf for tracking the number of distinct elements in suffixes).

2. Sets: A set is used to keep track of distinct elements because sets naturally eliminate duplicates.

The implementation follows these steps:

• Initialize the suf list to have n + 1 zeros, where n is the length of nums. This is done to handle the suffix count of distinct elements as well as accounting for the empty subarray at the end.

• Start by populating the suf array from the end of nums moving towards the beginning. For each index i, add nums[i] to a set (this ensures only distinct elements are counted). Then, set suf[i] to the current size of the set, which represents the count of distinct elements from nums[i] to the end of the array.

• Once the suf array is complete, reset (clear) the set to use it for counting distinct elements in prefixes.

• Create an answer list ans initialized with zeros of length n to hold the final result.

• Iterate through nums from start to end. For each index i, add nums[i] to the set (again, ensuring uniqueness) and subtract the number of distinct elements in the suffix (obtained from suf[i + 1]) from the number of distinct elements in the prefix (the current size of the set). This gives the diff[i] value.

• Assign this result to ans[i], building the answer as the loop progresses.

• Finally, return the ans list once the loop is complete.

The key algorithmic insight here is to use two passes, one for suffixes and one for prefixes, with the help of sets to dynamically maintain the count of distinct elements efficiently. This negates the need to re-calculate the number of distinct elements for each index from scratch, thus improving the time complexity significantly.

Here's the main loop in the code explained:

for i in range(n - 1, -1, -1):
    s.add(nums[i])
    suf[i] = len(s)

s.clear()
for i, x in enumerate(nums):
    s.add(x)
    ans[i] = len(s) - suf[i + 1]

In the first loop, we fill the suf array with the count of distinct elements for the suffix starting from each index. In the second loop, we use the suf array and the prefix count of distinct elements to calculate the diff for each index, which is stored in ans.

Example Walkthrough

Let's illustrate the solution approach using a small example. Suppose we have the following integer array nums:

nums = [4, 6, 4, 3]

We need to calculate the diff array where diff[i] is the count of distinct elements from the start of nums up to i, minus the count of distinct elements from i+1 to the end of nums. Here's how we would apply the solution step by step:

• Initialize the suf list to length n+1 with zero values. n is the length of nums, so suf initially will be [0, 0, 0, 0, 0].
• Initialize an empty set s to record distinct elements as we go.

We first populate the suf array in backward fashion:

• For i = 3, nums[3] = 3. We add this to the set s, which becomes {3}, and suf[3] = 1.
• For i = 2, nums[2] = 4. We add this to the set s, which becomes {3, 4}, and suf[2] = 2.
• For i = 1, nums[1] = 6. We add this to the set s, which becomes {3, 4, 6}, and suf[1] = 3.
• For i = 0, nums[0] = 4. The set s is already {3, 4, 6}, so adding 4 doesn't change it, and suf[0] remains at 3.

Now our suf array looks like this: [3, 3, 2, 1, 0].

Next, we'll go over nums from left to right to build our diff array:

• Clear the set s.
• For i = 0, we add nums[0] = 4 to s, which becomes {4}. We calculate diff[0] = len(s) - suf[1] = 1 - 3 = -2.
• For i = 1, we add nums[1] = 6 to s, which becomes {4, 6}. We calculate diff[1] = len(s) - suf[2] = 2 - 2 = 0.
• For i = 2, we add nums[2] = 4 to s, which doesn't change it as 4 is already in the set. Thus, we calculate diff[2] = len(s) - suf[3] = 2 - 1 = 1.
• For i = 3, we add nums[3] = 3 to s, which becomes {3, 4, 6}. We calculate diff[3] = len(s) - suf[4] = 3 - 0 = 3.

Our final diff array is [-2, 0, 1, 3] which is the result of our algorithm.

Solution Implementation

Time and Space Complexity

The distinctDifferenceArray function aims to calculate a list where each element at index i reflects the count of unique numbers from index i to the end of the list nums, minus the count of unique numbers from index i+1 to the end.

Time Complexity

The function comprises two main loops which both iterate over the array nums:

1. The first loop runs backward from n-1 to 0. In each iteration, it adds the current element into the set s. The length of the set is then recorded in the suf array. Since adding an element to the set and checking its length are O(1) operations, this loop has a complexity of O(n), where n is the number of elements in nums.

2. The second loop runs forward from 0 to n. Similar to the first loop, elements are added to a cleared set s, and the length difference between the sets is stored in the ans array. The complexity of this loop is also O(n).

There are no nested loops, so the overall time complexity is the sum of the two loops, which is O(n) + O(n) = O(2n). Simplified, the time complexity is O(n).

Space Complexity

The space complexity includes the memory used by the data structures that scale with the input size:

1. The two sets, s and suf: s will contain at most n unique elements, thus O(n) space complexity. suf is an array of size n+1, which also gives O(n).

2. The ans array: Created to store the resulting differences and is of size n, contributing another O(n) space complexity.

3. Auxiliary variables like i, x, and n: These use constant space, O(1).

Summing these up, the total space complexity is O(n) + O(n) + O(n) which simplifies to O(3n). However, in Big O notation, constant factors are discarded, so the space complexity simplifies to O(n).

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