Sliding window technique in Data Structures Theory - Time & Space Complexity

Time Complexity: Sliding window technique

O(n)

Understanding Time Complexity

Analyzing time complexity helps us see how fast the sliding window technique works as input grows.

We want to know how the number of steps changes when the input size increases.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function maxSumSubarray(arr, k) {
  let maxSum = 0, windowSum = 0;
  for (let i = 0; i < k; i++) {
    windowSum += arr[i];
  }
  maxSum = windowSum;
  for (let end = k; end < arr.length; end++) {
    windowSum += arr[end] - arr[end - k];
    if (windowSum > maxSum) maxSum = windowSum;
  }
  return maxSum;
}
    

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

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

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

How Execution Grows With Input

As the array gets bigger, 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 input roughly doubles the work done.

Final Time Complexity

Time Complexity: O(n)

This means the time to complete grows in direct proportion to the size of the input array.

Common Mistake

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

[OK] Correct: The sliding window moves the window forward by one step each time without redoing all work, so it only needs one main loop, making it much faster.

Interview Connect

Understanding sliding window time complexity shows you can spot efficient ways to handle sequences, a skill valued in many coding challenges.

Self-Check

"What if we needed to find the maximum sum of any subarray but the size k could change each time? How would the time complexity change?"

Practice

1. What is the main purpose of the sliding window technique in data processing?

easy

A. To efficiently process continuous segments of data without repeated work

B. To sort data in ascending order

C. To store data permanently in memory

D. To randomly access elements in a list

Solution

1. Step 1: Understand the sliding window concept

   The sliding window technique moves a fixed-size window over data to process parts efficiently.

2. Step 2: Identify the main goal

   It avoids repeated calculations by updating results as the window moves, improving efficiency.

3. Final Answer:

   To efficiently process continuous segments of data without repeated work -> Option A

4. Quick Check:

   Sliding window = efficient continuous data processing [OK]

Hint: Think of a moving window checking parts step-by-step [OK]

Common Mistakes:

• Confusing sliding window with sorting algorithms
• Thinking it stores data permanently
• Assuming it accesses data randomly

2. Which of the following is the correct way to initialize a sliding window of size 3 over a list arr in Python?

easy

A. window = arr{0:3}

B. window = arr(0,3)

C. window = arr[0:3]

D. window = arr[3]

Solution

1. Step 1: Recall Python slicing syntax

   Python uses square brackets and colon to slice lists: arr[start:end] extracts elements from start up to but not including end.

2. Step 2: Apply slicing for window size 3

   To get first 3 elements, use arr[0:3]. Other options use incorrect syntax.

3. Final Answer:

   window = arr[0:3] -> Option C

4. Quick Check:

   Python slice for first 3 elements = arr[0:3] [OK]

Hint: Remember Python slices use square brackets and colon [OK]

Common Mistakes:

• Using parentheses instead of brackets for slicing
• Using curly braces which are invalid for lists
• Accessing a single element instead of a slice

3. Consider the Python code below using sliding window to find the maximum sum of any 3 consecutive numbers in arr:

arr = [2, 1, 5, 1, 3, 2]
window_sum = sum(arr[0:3])
max_sum = window_sum
for i in range(3, len(arr)):
    window_sum += arr[i] - arr[i-3]
    if window_sum > max_sum:
        max_sum = window_sum
print(max_sum)

What is the output?

medium

A. 7

B. 8

C. 6

D. 9

Solution

1. Step 1: Calculate initial window sum

   Sum of first 3 elements: 2 + 1 + 5 = 8. Set max_sum = 8.

2. Step 2: Slide window and update sums

   i=3: window_sum = 8 + arr[3] - arr[0] = 8 + 1 - 2 = 7 (max_sum=8)
   i=4: window_sum = 7 + arr[4] - arr[1] = 7 + 3 - 1 = 9 (max_sum=9)
   i=5: window_sum = 9 + arr[5] - arr[2] = 9 + 2 - 5 = 6 (max_sum=9)

3. Final Answer:

   9 -> Option D

4. Quick Check:

   Max sum of 3 consecutive = 9 [OK]

Hint: Add new element, subtract old element to update sum [OK]

Common Mistakes:

• Recalculating sum of window each time instead of updating
• Mixing up indices when subtracting old element
• Not updating max_sum correctly

4. The following code attempts to find the minimum sum of any 2 consecutive numbers in arr using sliding window:

arr = [4, 3, 7, 1, 2]
window_sum = arr[0] + arr[1]
min_sum = window_sum
for i in range(2, len(arr)):
    window_sum = window_sum + arr[i] - arr[i-1]
    if window_sum < min_sum:
        min_sum = window_sum
print(min_sum)

What is the error in this code?

medium

A. The window size is incorrectly handled in subtraction index

B. The loop range should start from 1, not 2

C. The initial window sum is calculated incorrectly

D. The comparison operator in if statement is wrong

Solution

1. Step 1: Analyze window size and indices

   The window size is 2. When sliding to include arr[i], subtract the leaving element arr[i-2].

2. Step 2: Identify the error

   Code subtracts arr[i-1], but should subtract arr[i-2]. arr[i-1] remains in the window.

3. Step 3: Verify with trace

   i=2: should be 7 + 7 - 4 = 10, but code 7 + 7 - 3 = 11 (wrong).

4. Final Answer:

   The window size is incorrectly handled in subtraction index -> Option A

5. Quick Check:

   For size k=2, subtract arr[i-k] [OK]

Hint: Check subtraction index matches window size [OK]

Common Mistakes:

• Using wrong index for element leaving the window
• Starting loop at wrong index
• Incorrect initial sum calculation

5. You want to find the longest substring of a string s that contains at most 2 distinct characters using the sliding window technique. Which approach correctly applies the sliding window here?

hard

A. Fix window size to 2 and slide over the string

B. Expand the window by moving right pointer, shrink from left when distinct chars exceed 2

C. Count total distinct characters in the string once and return that count

D. Use nested loops to check all substrings of length 2

Solution

1. Step 1: Understand problem requirements

   We want the longest substring with at most 2 distinct characters, so window size is variable.

2. Step 2: Apply sliding window logic

   Expand window by moving right pointer. If distinct chars exceed 2, shrink window from left until condition holds.

3. Step 3: Evaluate options

   Expand the window by moving right pointer, shrink from left when distinct chars exceed 2 matches this approach. Fix window size to 2 and slide over the string fixes window size incorrectly. Options C and D do not use sliding window properly.

4. Final Answer:

   Expand the window by moving right pointer, shrink from left when distinct chars exceed 2 -> Option B

5. Quick Check:

   Variable window size with condition check = Expand the window by moving right pointer, shrink from left when distinct chars exceed 2 [OK]

Hint: Adjust window size dynamically based on distinct chars count [OK]

Common Mistakes:

• Fixing window size instead of adjusting dynamically
• Not shrinking window when condition breaks
• Using brute force nested loops instead of sliding window