algoadvance

You are given an array of integers nums. You need to return the sum of the squares of all the distinct elements in all possible subarrays of nums.

Clarifying Questions

1. What are the constraints on the length of the array, and the range of the integer values within it?
   • Let’s assume 1 <= len(nums) <= 1000 and -1000 <= nums[i] <= 1000.
2. What should be the result if the array is empty?
   • Given the constraint 1 <= len(nums), the array will not be empty.
3. Should the subarrays be considered with distinct elements only, or are all subarrays considered, but computation includes distinction?
   • All subarrays are considered, but we compute the sum of squares of their distinct elements.

Strategy

To solve this problem, we’ll:

1. Generate all subarrays of nums.
2. For each subarray, extract unique elements.
3. Compute the sum of the squares of these unique elements.
4. Accumulate the results for the final output.

This approach ensures we compute the contribution by the distinct elements in all subarrays.

Code

Let’s implement this step-by-step.

def sum_of_squares_of_distinct_elements(nums):
    n = len(nums)
    total_sum = 0
    
    for i in range(n):
        unique_elements = set()
        for j in range(i, n):
            unique_elements.add(nums[j])
            total_sum += sum(x ** 2 for x in unique_elements)
    
    return total_sum

# Example Usage:
nums = [1, 2, 1]
print(sum_of_squares_of_distinct_elements(nums))  # Output will depend on the summing up of squared distinct elements in all subarrays

Time Complexity

Let’s analyze the time complexity:

• Generating Subarrays: The nested loop will generate all subarrays and each element can appear in subarrays n * (n + 1) / 2 times.

• Set Operations: Inserting elements into a set has an average time complexity of (O(1)).

• Sum and Square Computation: For each subarray, computing the sum of squares of unique elements takes (O(k)) time where (k) is the number of unique elements in the subarray.

In the worst case (all elements being unique), the time complexity would roughly be (O(n^3)) since we are summing over a nested loop which itself goes up to n more operations per element.

Thus, Time Complexity: (O(n^3))

This approach should be reasonable given the constraints stated.

Further Considerations

We could possibly optimize by exploring more efficient summation techniques or leveraging data structures that dynamically keep track of elements’ contributions to reduce repeated calculations, but this basic approach fits within reasonable constraint limits.

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