To solve the problem of finding the maximum sum of a contiguous subarray of fixed length (k), we use the sliding window technique—an efficient approach that avoids redundant calculations and runs in (O(n)) time.

Approach

The sliding window technique works by maintaining a "window" of size (k) over the array. Instead of recalculating the sum of the entire window each time it slides, we adjust the sum incrementally:

1. Initial Window: Compute the sum of the first (k) elements.
2. Slide the Window: For each subsequent element:
   • Subtract the element that is leaving the window (leftmost element of the previous window).
   • Add the new element entering the window (next element to the right of the previous window).
3. Track Maximum: Keep updating the maximum sum whenever the new window sum is larger than the current maximum.

Solution Code

def max_subarray_sum(arr, k):
    # Handle edge cases: if array length < k or k <= 0
    if len(arr) < k or k <= 0:
        return None  # Or raise an error based on requirements

    # Calculate initial window sum
    current_sum = sum(arr[:k])
    max_sum = current_sum

    # Slide the window across the array
    for i in range(k, len(arr)):
        current_sum += arr[i] - arr[i - k]
        if current_sum > max_sum:
            max_sum = current_sum

    return max_sum

Explanation

Let’s take an example to illustrate:

• Array: [2,1,5,1,3,2], (k=3)
• Initial Window: Sum of first 3 elements → (2+1+5=8) (max=8).
• Next Window: Subtract 2 (leaving) and add 1 (entering) → (1+5+1=7) (max remains 8).
• Next Window: Subtract1 and add3 → (5+1+3=9) (max=9).
• Next Window: Subtract5 and add2 → (1+3+2=6) (max remains9).

The final maximum sum is 9.

Key Advantages

• Time Complexity: (O(n)) (each element is processed once).
• Space Complexity: (O(1)) (only a few variables are used, no extra space for the window).

This approach works for arrays with both positive and negative numbers, as it correctly tracks the maximum sum of any valid window. For edge cases like (k=1) (max element) or (k=\text{length of array}) (sum of all elements), the solution handles them seamlessly.

Answer: The maximum sum of the subarray of size (k) is computed using the sliding window technique as shown above. For example, if the input array is [2,1,5,1,3,2] and (k=3), the result is 9. The code returns the correct maximum sum for any valid input.

\boxed{9} (for the example given; replace with actual result based on input array and k)

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