Introduction

The Maximum Sum Subarray of Size K problem involves finding the contiguous subarray of size k that has the largest sum.

Given an array arr[] of size n and an integer k, the task is to find the maximum sum among all possible subarrays of size k.

This problem is one of the most important beginner-level Sliding Window problems and helps in understanding window-based optimization techniques.

Example

Input: arr[] = [2, 1, 5, 1, 3, 2]k = 3
Output: 9
Explanation:
Subarray [2, 1, 5] → Sum = 8
Subarray [1, 5, 1] → Sum = 7
Subarray [5, 1, 3] → Sum = 9
Subarray [1, 3, 2] → Sum = 6
Maximum Sum = 9

Input: arr[] = [4, 2, 1, 7, 8, 1]
k = 2
Output: 15
Explanation:
Subarray [4, 2] → Sum = 6
Subarray [2, 1] → Sum = 3
Subarray [1, 7] → Sum = 8
Subarray [7, 8] → Sum = 15
Subarray [8, 1] → Sum = 9
Maximum Sum = 15

Constraints

   1 <= n <= 10^5
      1 <= k <= n
-10^9 <= arr[i] <= 10^9

Approach 1 : Brute Force

Explanation

The simplest way to solve this problem is:

  1. Generate every possible subarray of size k
  2. Calculate the sum of each subarray
  3. Track the maximum sum

This approach is easy to understand but repeatedly calculating sums increases time complexity.

Steps

  1. Traverse the array from index 0 to n-k;
  2. For every starting index:
    • calculate sum of next k elements
  3. Compare the sum with maximum sum.
  4. Return the final maximum sum.

Dry Run

Input Array: [2, 1, 5, 1, 3, 2]k = 3
Subarray [2, 1, 5]:
Sum = 8
Maximum Sum = 8
Subarray [1, 5, 1]:
Sum = 7
Maximum remains 8
Subarray [5, 1, 3]:
Sum = 9
Update Maximum Sum = 9
Subarray [1, 3, 2]:
Sum = 6
Maximum remains 9
Final Result: 9

Brute Force Code 

Complexity Analysis

Time Complexity: O(n * k)Explanation:
For every subarray, k elements are traversed again.

Space Complexity: O(1)
Explanation:
No extra data structures are used.

Approach 2 : Optimized Solution (Sliding Window)

Explanation

Instead of recalculating the sum for every subarray, we can use the Sliding Window Technique.

The idea is:

  1. Calculate the first window sum
  2. Slide the window forward
  3. Add the new element entering the window
  4. Remove the old element leaving the window

This avoids repeated calculations and improves efficiency.

Steps

  1. Calculate sum of first k elements.
  2. Store it as current maximum sum.
  3. Slide the window one step at a time:
    • subtract outgoing element
    • add incoming element
  4. Update maximum sum whenever needed.
  5. Return maximum sum.
Dry Run
Input Array: [2, 1, 5, 1, 3, 2]
k = 3
First Window [2, 1, 5]:
Window Sum = 8
Maximum Sum = 8
Slide Window:
Remove 2
Add 1
New Window [1, 5, 1]
Window Sum = 7
Maximum remains 8
Slide Window Again:
Remove 1
Add 3
New Window [5, 1, 3]
Window Sum = 9
Update Maximum Sum = 9
Slide Window Again:
Remove 5
Add 2
New Window [1, 3, 2]
Window Sum = 6
Maximum remains 9
Final Result: 9

Optimized Code 

Complexity Analysis

Time Complexity: O(n)Explanation:Each element is added and removed from the window exactly once.


Space Complexity: O(1)
Explanation:
Each element is added and removed from the window exactly once.

Edge Cases

  1. Array contains negative numbers
  2. Window size equals array size
  3. Window size is 1
  4. Array contains all same elements

Why This Problem is Important

This problem helps in understanding:

  1. Sliding Window Technique
  2. Window optimization
  3. Efficient subarray processing
  4. Traversal optimization
  5. Sum-based problems

It is one of the most important Sliding Window interview problems.

Real-World Applications

Sliding Window techniques are used in:

  1. Network traffic analysis
  2. Financial data processing
  3. Streaming systems
  4. Sensor data analysis
  5. Time-series processing

Common Mistakes

  1. Incorrect window boundaries
  2. Forgetting outgoing element removal
  3. Incorrect initialization of first window
  4. Traversing beyond array size

Interview Tips

Interviewers often expect:

  1. Sliding Window optimization
  2. O(n) solution
  3. Proper window movement logic

Always explain why recalculating the entire window sum is inefficient.

Related Questions

  1. First Negative in Window
  2. Longest Subarray with Sum K
  3. Maximum Average Subarray
  4. Minimum Size Subarray Sum
  5. Sliding Window Maximum

Final Takeaway

The Maximum Sum Subarray of Size K problem is a fundamental Sliding Window problem that teaches efficient subarray processing and optimization techniques. Understanding this problem builds a strong foundation for advanced Sliding Window and Two Pointer problems.