Problem Description

The task is to calculate a particular sum related to all subarrays of a given array nums that is 0-indexed (meaning that the indices of the elements start at 0). Specifically, you must find the sum of the squares of the "distinct counts" of all subarrays of nums. A "distinct count" refers to the number of unique elements present in a subarray.

A subarray is defined as any contiguous sequence of elements from the array. Consequently, for an array of length n, there are n * (n + 1) / 2 possible subarrays since each element can be the starting point of a subarray and can extend to any of the remaining elements, including itself.

To clarify, consider a subarray nums[i..j] where i and j represent the starting and ending indices, respectively, and adhere to 0 <= i <= j < nums.length. The distinct count of such a subarray is the count of unique values within it.

The problem asks for the sum of the squares of each subarray's distinct count. This means that for each subarray, we count how many different numbers it has, square this count, and then add all these squares together to get the final answer.

Intuition

To solve this problem, a straightforward approach is to look at each possible subarray, calculate its distinct count, square it, and then sum these values.

We start by iterating over all possible starting points of subarrays. For each starting point, we create a new subarray that begins at this index. We then incrementally add elements to the end of this subarray one by one, extending it until we reach the end of the array.

As we add each new element to our current subarray, we maintain a set that holds the distinct elements found so far. This set allows us to easily keep track of the count of unique elements, as adding a duplicate to a set does not change its size. The size of this set, squared, represents the contribution of the current subarray to the overall answer.

So the overall process is:

1. Enumerate the starting point i of each subarray.
2. For each start point, iterate through the array to extend the subarray until the end of the array, each time keeping track of unique elements using a set.
3. With each addition to the subarray, calculate the size of the set (the distinct count), square it, and add this to a running total.
4. Once we have done this for all subarrays, return the total sum as the answer.

This approach ensures that we consider each subarray exactly once and correctly calculate the distinct count for it, which leads us steadily to the answer.

Solution Approach

To implement the solution for the above-defined problem, the following steps solidify the approach using the Python programming language.

• We initiate by establishing a class named Solution which contains the function sumCounts(self, nums: List[int]) -> int to execute the solution logic.

• We define a variable ans to store our accumulated result and start it with a value of 0. Similarly, we assign n to the length of nums to keep track of the array's size, which we will repeatedly use in our iterations.

• We traverse each element in the array using a for loop, where our iterator i goes from 0 up to, but not including, n. This iterator signifies the beginning of each subarray we are going to evaluate.

• Inside this loop, we instantiate an empty set s. This will hold the distinct elements of the current subarray originating from index i.

• We then nest another for loop and set our second iterator j to range from i to n. This loop will consider every possible endpoint for the subarray starting at i. In other words, we are examining every subarray nums[i..j].

• We add the current element nums[j] into our set s. The unique property of a set ensures that it will only contain distinct elements.

• Finally, we calculate the contribution of the current subarray to our answer by taking the size of the set (our distinct count) and squaring it, leading to len(s) * len(s).

• That squared value is then added to ans, which is incrementally growing to encompass the sum of the squares of distinct counts of all subarrays considered so far.

• Once we have completed the enumeration of all subarrays and their contributions, we return ans as the final sum.

In terms of data structures and algorithms, this solution leverages a set to efficiently track unique elements, and it uses nested loops to enumerate all possible subarrays. The algorithm's runtime would depend on the number of subarrays it has to consider, which is on the order of O(n^2), as for each starting index i, we could potentially look at n-i endpoints.

Summarized, the implementation is direct and makes use of simple data structures to deliver the desired outcome.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have an array nums with the elements [1,2,1]. We will walk through the algorithm step by step to calculate the sum of the squares of the distinct counts of all subarrays.

1. Initialize variables:

   • ans is initialized to 0.
   • n is assigned the value 3 (since the array has 3 elements).

2. Iterate over each possible start index i of subarrays:

   • For i = 0: Initialize s as an empty set.
     • For j = 0: Add nums[j] (which is 1) to the set. Now, s = {1}. The distinct count is 1 with a square of 1. Add this to ans, so ans = 1.
     • For j = 1: Add nums[j] (which is 2) to the set. Now, s = {1, 2}. The distinct count is 2 with a square of 4. Add this to ans, so ans = 1 + 4 = 5.
     • For j = 2: Add nums[j] (which is 1) to the set. However, 1 is already in the set, so s = {1, 2}. The distinct count remains 2 with a square of 4. Add this to ans, so ans = 5 + 4 = 9.
   • For i = 1: Reset s as an empty set.
     • For j = 1: Add nums[j] (which is 2) to the set. Now, s = {2}. The distinct count is 1 with a square of 1. Add this to ans, so ans = 9 + 1 = 10.
     • For j = 2: Add nums[j] (which is 1) to the set. Now, s = {1, 2}. The distinct count is 2 with a square of 4. Add this to ans, so ans = 10 + 4 = 14.
   • For i = 2: Reset s as an empty set.
     • For j = 2: Add nums[j] (which is 1) to the set. Now, s = {1}. The distinct count is 1 with a square of 1. Add this to ans, so ans = 14 + 1 = 15.

3. Final Result: After considering all subarrays, the sum of the squares of distinct counts is equal to ans = 15.

Thus, following the solution approach, we've processed each subarray of nums = [1,2,1], computed the square of the distinct count for each subarray, and added them together to produce the final result. The sets formed for each subarray were helpful in tracking the unique elements, and the nested for loops made sure that we considered every possible subarray.

Solution Implementation

Time and Space Complexity

Time Complexity

The provided code consists of two nested loops: the outer loop goes through each element in the input list nums, and the inner loop iterates over every subsequent element, adding them to a set s. The inner operation in which the set size len(s) is squared and added to the running ans has a constant time complexity since finding the length of a set and performing arithmetic operations are both done in constant time.

The time complexity can be understood as follows. For each index i, the inner loop runs n-i times, where n is the length of nums. So, the total number of iterations of the inner loop is n + (n-1) + (n-2) + ... + 1, which simplifies to n*(n+1)/2. Since each iteration involves a constant-time set addition and calculation, this algorithm is quadratic in nature. Using Big O notation, we write the time complexity as O(n^2).

Space Complexity

The space complexity is determined by the additional space used by the algorithm independent of the input size. In this case, the set s is the main factor contributing to space complexity. In the worst-case scenario, the set can grow to include all unique elements of nums. Since we do not use any other data structure that scales with the size of the input, the space complexity is O(n), where n is the length of the array nums.