Sliding Window on Arrays in DSA C - Time & Space Complexity

Time Complexity: Sliding Window on Arrays

O(n)

Understanding Time Complexity

We want to understand how the time needed changes when using the sliding window technique on arrays.

How does the number of steps grow as the array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int maxSumSubarray(int arr[], int n, int k) {
    int max_sum = 0, window_sum = 0;
    for (int i = 0; i < k; i++) {
        window_sum += arr[i];
    }
    max_sum = window_sum;
    for (int i = k; i < n; i++) {
        window_sum += arr[i] - arr[i - k];
        if (window_sum > max_sum) {
            max_sum = window_sum;
        }
    }
    return max_sum;
}
    

This code finds the maximum sum of any subarray of size k using a sliding window.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Two loops that each run over parts of the array.
• How many times: First loop runs k times, second loop runs (n - k) times.

How Execution Grows With Input

The total steps grow roughly in a straight line with the size of the array.

Input Size (n)	Approx. Operations
10	About 10 steps
100	About 100 steps
1000	About 1000 steps

Pattern observation: Doubling the array size roughly doubles the work done.

Final Time Complexity

Time Complexity: O(n)

This means the time needed grows in a straight line with the size of the input array.

Common Mistake

[X] Wrong: "The sliding window has nested loops, so it must be O(n²)."

[OK] Correct: The loops run one after another, not inside each other, so total steps add up, not multiply.

Interview Connect

Understanding sliding window time complexity helps you explain efficient solutions clearly and confidently.

Self-Check

"What if we changed the window size k to vary with n, like k = n/2? How would the time complexity change?"

Practice

1. What is the main purpose of using a sliding window technique on arrays?

easy

A. To efficiently process subarrays of fixed size without recomputing everything

B. To sort the array in ascending order

C. To reverse the array elements

D. To find the maximum element in the entire array

Solution

1. Step 1: Understand sliding window concept

   The sliding window technique moves a fixed-size window over the array to process subarrays efficiently.

2. Step 2: Identify the main use

   It avoids recomputing sums or other values for each subarray by updating results incrementally.

3. Final Answer:

   To efficiently process subarrays of fixed size without recomputing everything -> Option A

4. Quick Check:

   Sliding window = efficient subarray processing [OK]

Hint: Sliding window avoids repeated work on overlapping subarrays [OK]

Common Mistakes:

• Thinking it sorts the array
• Assuming it reverses elements
• Confusing it with finding max of whole array

2. Which of the following is the correct syntax to initialize a sliding window sum for the first k elements in C?

easy

A. int sum = 0; for(int i = 1; i <= k; i++) sum += arr[i];

B. int sum = arr[0] + arr[k];

C. int sum = 0; for(int i = 0; i < k; i++) sum += arr[i];

D. int sum = arr[k]; for(int i = 0; i < k; i++) sum += arr[i];

Solution

1. Step 1: Check loop range for first k elements

   The loop should start at 0 and run while i < k to cover first k elements.

2. Step 2: Verify sum initialization and accumulation

   Sum starts at 0 and adds arr[i] for each i in 0 to k-1.

3. Final Answer:

   int sum = 0; for(int i = 0; i < k; i++) sum += arr[i]; -> Option C

4. Quick Check:

   Loop 0 to k-1 sums first k elements [OK]

Hint: Loop from 0 to k-1 to sum first k elements [OK]

Common Mistakes:

• Starting loop at 1 instead of 0
• Adding arr[k] which is outside first k elements
• Initializing sum incorrectly

3. What is the output of the following C code snippet?

int arr[] = {1, 3, 2, 6, -1, 4, 1, 8, 2};
int k = 5;
int max_sum = 0, window_sum = 0;
for(int i = 0; i < k; i++) window_sum += arr[i];
max_sum = window_sum;
for(int i = k; i < 9; i++) {
    window_sum += arr[i] - arr[i - k];
    if(window_sum > max_sum) max_sum = window_sum;
}
printf("%d", max_sum);

medium

A. 18

B. 16

C. 17

D. 15

Solution

1. Step 1: Calculate initial window sum of first 5 elements

   Sum = 1+3+2+6+(-1) = 11

2. Step 2: Slide window and update sums

   Next windows:
   i=5: sum=11 + 4 - 1 = 14
   i=6: sum=14 + 1 - 3 = 12
   i=7: sum=12 + 8 - 2 = 18
   i=8: sum=18 + 2 - 6 = 14
   Max sum found = 18

3. Final Answer:

   18 -> Option A

4. Quick Check:

   Max sliding window sum = 18 [OK]

Hint: Update sum by adding new and subtracting old element [OK]

Common Mistakes:

• Not subtracting the element leaving the window
• Wrong initial sum calculation
• Off-by-one errors in loop

4. Identify the error in the following sliding window code snippet and choose the correct fix:

int arr[] = {2, 1, 5, 1, 3, 2};
int k = 3;
int window_sum = 0;
for(int i = 0; i <= k; i++) window_sum += arr[i];
for(int i = k; i < 6; i++) {
    window_sum += arr[i] - arr[i - k];
    printf("%d ", window_sum);
}

medium

A. No error, code is correct

B. Change loop condition to i < k in second for loop

C. Initialize window_sum to arr[0]

D. Change loop condition to i < k in first for loop

Solution

1. Step 1: Check first loop range for initial sum

   Loop runs i = 0 to i = k (inclusive), so 4 elements summed instead of 3.

2. Step 2: Fix loop condition to sum exactly k elements

   Change condition to i < k to sum first 3 elements only.

3. Final Answer:

   Change loop condition to i < k in first for loop -> Option D

4. Quick Check:

   Initial sum must cover exactly k elements [OK]

Hint: Use i < k to sum first k elements, not i <= k [OK]

Common Mistakes:

• Using <= instead of < in loop condition
• Changing wrong loop's condition
• Assuming initialization error

5. Given an array of integers and a window size k, how would you find the minimum sum of any contiguous subarray of size k using the sliding window technique?

hard

A. Sort the array and pick first k elements to sum

B. Initialize sum of first k elements, slide window updating sum and track minimum sum found

C. Use nested loops to check all subarrays and find minimum sum

D. Reverse the array and sum last k elements

Solution

1. Step 1: Calculate initial sum of first k elements

   Sum the first k elements to start the sliding window.

2. Step 2: Slide window across array updating sum and track minimum

   For each new element, add it and subtract the element leaving the window, update minimum sum if current sum is smaller.

3. Final Answer:

   Initialize sum of first k elements, slide window updating sum and track minimum sum found -> Option B

4. Quick Check:

   Sliding window tracks sums efficiently to find minimum [OK]

Hint: Track min sum by updating window sum while sliding [OK]

Common Mistakes:

• Sorting array instead of sliding window
• Using nested loops causing inefficiency
• Ignoring elements leaving the window