Solution Explanation for Range Addition

This problem can be efficiently solved using the concept of difference arrays. A difference array transforms the problem of adding a value to a range of elements into a problem of updating only two elements. This significantly reduces the time complexity compared to iterating through the entire range for each update.

Approach:

1. Difference Array: Create a difference array d of the same length as the input array arr. d[i] will store the difference between arr[i] and arr[i-1] (where arr[-1] is considered 0).

2. Update the Difference Array: For each update [startIdx, endIdx, inc], we do the following:

   • Add inc to d[startIdx]. This accounts for the increase at the starting index.
   • Subtract inc from d[endIdx + 1]. This accounts for the decrease needed to prevent the increase from affecting elements beyond endIdx.

3. Prefix Sum: Calculate the prefix sum of the difference array d. The resulting prefix sum array represents the modified array arr. This is because the prefix sum at each index accumulates the differences from the beginning, effectively reconstructing the original array after all updates.

Time Complexity Analysis:

• Creating the difference array: O(n), where n is the length of the input array.
• Updating the difference array: O(m), where m is the number of updates. Each update takes constant time.
• Calculating the prefix sum: O(n).

Therefore, the overall time complexity is O(n + m), which is linear in terms of the array length and the number of updates.

Space Complexity Analysis:

The space complexity is O(n) because we need to store the difference array and the final modified array (although in some implementations, we can modify the input array in place).

Code Examples:

The code examples provided demonstrate the difference array approach in several programming languages. They follow these steps:

1. Initialize a difference array with zeros.
2. Iterate through the updates, updating the difference array as described above.
3. Compute the prefix sum of the difference array to get the final modified array.

The Binary Indexed Tree (BIT) solution is also provided, offering a more optimized solution with a time complexity of O(m log n). While more complex to understand and implement, it can be advantageous when dealing with a very large number of updates. This approach uses BIT to efficiently handle the cumulative updates.

Remember to choose the solution that best fits the constraints and performance requirements of your specific problem instance. For most cases with moderate n and m, the difference array method is sufficiently efficient and easier to implement.