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Sum of Variable Length Subarrays

Difficulty: Easy

Problem Description

You are given an integer array nums of size n. For each index i where 0 <= i < n, define a subarray nums[start ... i] where start = max(0, i - nums[i]). Return the total sum of all elements from the subarray defined for each index in the array.

Key Insights

  • Each index i contributes to a subarray that starts from start = max(0, i - nums[i]) to i.
  • The sum of the elements in the subarray can be computed for each index, and then all these sums need to be aggregated.
  • A direct approach would involve iterating through the array and calculating the sum for each subarray, but we can optimize this using prefix sums.

Space and Time Complexity

Time Complexity: O(n^2)
Space Complexity: O(1) (if we do not count the input storage)

Solution

To solve the problem, we can iterate through each index of the array and calculate the sum of the relevant subarray by determining its starting point using start = max(0, i - nums[i]). For each index, we will sum the elements from start to i and accumulate this sum into a total sum variable.

We can also optimize the calculation of the subarray sum using a prefix sum approach if needed, but given the constraints, a straightforward implementation will work efficiently.

Code Solutions

def sum_of_variable_length_subarrays(nums):
    total_sum = 0
    n = len(nums)
    
    for i in range(n):
        start = max(0, i - nums[i])  # Calculate the starting index of the subarray
        total_sum += sum(nums[start:i+1])  # Sum the elements in the subarray from start to i
    
    return total_sum
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