Prefix Sum Array in DSA C# - Time & Space Complexity

Time Complexity: Prefix Sum Array

O(n)

Understanding Time Complexity

We want to understand how fast we can build a prefix sum array from a list of numbers.

How does the time to create this array grow as the input list gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int[] BuildPrefixSum(int[] nums) {
    int n = nums.Length;
    int[] prefixSum = new int[n];
    prefixSum[0] = nums[0];
    for (int i = 1; i < n; i++) {
        prefixSum[i] = prefixSum[i - 1] + nums[i];
    }
    return prefixSum;
}
    

This code creates a new array where each element is the sum of all numbers up to that position in the original array.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The for-loop that adds the current number to the sum of all previous numbers.
• How many times: This loop runs once for each element after the first, so (n - 1) times.

How Execution Grows With Input

As the input size grows, the number of additions grows roughly the same way.

Input Size (n)	Approx. Operations
10	9 additions
100	99 additions
1000	999 additions

Pattern observation: The operations increase directly with the input size, almost one addition per element.

Final Time Complexity

Time Complexity: O(n)

This means the time to build the prefix sum grows in a straight line with the size of the input array.

Common Mistake

[X] Wrong: "Since we add sums repeatedly, the time must be quadratic (like n squared)."

[OK] Correct: Each addition uses the previous sum, so we do only one addition per element, not a new sum from scratch each time.

Interview Connect

Knowing how prefix sums work and their time cost helps you answer questions about fast range queries and cumulative calculations.

Self-Check

"What if we changed the code to compute prefix sums recursively? How would the time complexity change?"

Practice

1. What is the main purpose of a Prefix Sum Array in algorithms?

easy

A. To quickly find the sum of elements in any subarray

B. To sort the array elements in ascending order

C. To find the maximum element in the array

D. To reverse the array elements

Solution

1. Step 1: Understand what prefix sums represent

   A prefix sum array stores cumulative sums of elements from the start up to each index.

2. Step 2: Identify the use of prefix sums

   Using prefix sums, we can find the sum of any subarray quickly by subtracting two prefix sums.

3. Final Answer:

   To quickly find the sum of elements in any subarray -> Option A

4. Quick Check:

   Prefix Sum Array = Fast subarray sums [OK]

Hint: Prefix sums help sum subarrays fast by subtraction [OK]

Common Mistakes:

• Thinking prefix sums sort the array
• Confusing prefix sums with maximum element search
• Assuming prefix sums reverse the array

2. Which of the following C# code snippets correctly initializes a prefix sum array prefix from an integer array arr?

easy

A. int[] prefix = new int[arr.Length+1]; for(int i=1; i<=arr.Length; i++) prefix[i] = prefix[i-1] + arr[i];

B. int[] prefix = new int[arr.Length]; for(int i=0; i

C. int[] prefix = new int[arr.Length]; prefix[0] = arr[0]; for(int i=1; i

D. int[] prefix = new int[arr.Length]; prefix[0] = 0; for(int i=1; i

Solution

1. Step 1: Check initialization of prefix array

   int[] prefix = new int[arr.Length]; prefix[0] = arr[0]; for(int i=1; i

2. Step 2: Verify loop for cumulative sum

   The loop starts from 1 and adds prefix[i-1] + arr[i], correctly building prefix sums.

3. Final Answer:

   int[] prefix = new int[arr.Length]; prefix[0] = arr[0]; for(int i=1; i -> Option C

4. Quick Check:

   Correct prefix sum initialization = int[] prefix = new int[arr.Length]; prefix[0] = arr[0]; for(int i=1; i

Hint: Start prefix[0] = arr[0], then add previous prefix + current element [OK]

Common Mistakes:

• Using wrong array length for prefix
• Starting loop from 0 instead of 1
• Accessing arr out of bounds (like arr[i] when i == arr.Length)

3. Given the array arr = new int[] {2, 4, 6, 8} and its prefix sum array prefix computed as usual, what is the output of prefix[3] - prefix[1]?

medium

A. 18

B. 10

C. 12

D. 14

Solution

1. Step 1: Compute prefix sum array

   prefix[0]=2, prefix[1]=2+4=6, prefix[2]=6+6=12, prefix[3]=12+8=20

2. Step 2: Calculate prefix[3] - prefix[1]

   20 - 6 = 14

3. Final Answer:

   14 -> Option D

4. Quick Check:

   Sum from index 2 to 3 = 6+8 = 14 [OK]

Hint: Subtract prefix sums to get subarray sum [OK]

Common Mistakes:

• Confusing indices and subtracting wrong prefix values
• Adding prefix sums instead of subtracting
• Forgetting prefix sums start at index 0

4. Consider the following code snippet to build a prefix sum array. Identify the error:

int[] prefix = new int[arr.Length];
prefix[0] = 0;
for(int i = 1; i < arr.Length; i++) {
    prefix[i] = prefix[i-1] + arr[i];
}

medium

A. Loop should start from i = 0, not 1

B. prefix[0] should be arr[0], not 0

C. Array prefix should have length arr.Length + 1

D. No error, code is correct

Solution

1. Step 1: Check initialization of prefix[0]

   Setting prefix[0] = 0 is incorrect; it should be arr[0] to start cumulative sum correctly.

2. Step 2: Verify loop and array length

   The loop starting at 1 and array length are fine; only prefix[0] initialization is wrong.

3. Final Answer:

   prefix[0] should be arr[0], not 0 -> Option B

4. Quick Check:

   Correct prefix start = arr[0] [OK]

Hint: Initialize prefix[0] with arr[0], not zero [OK]

Common Mistakes:

• Starting loop at 0 causing index errors
• Using wrong prefix array size
• Assuming zero initialization is always correct

5. You have an integer array arr of length 6: {3, 1, 4, 1, 5, 9}. You want to find the sum of elements from index 2 to 5 inclusive using a prefix sum array prefix where prefix[i] stores sum of elements from index 0 to i. Which expression correctly computes this sum?

hard

A. prefix[5] - prefix[1]

B. prefix[5] - prefix[2]

C. prefix[4] - prefix[1]

D. prefix[5] + prefix[1]

Solution

1. Step 1: Understand prefix sum indexing

   prefix[i] = sum of arr[0] to arr[i]. To get sum from index 2 to 5, subtract prefix[1] (sum up to index 1) from prefix[5].

2. Step 2: Write the correct expression

   Sum = prefix[5] - prefix[1]

3. Final Answer:

   prefix[5] - prefix[1] -> Option A

4. Quick Check:

   Sum(2 to 5) = prefix[5] - prefix[1] [OK]

Hint: Sum(i to j) = prefix[j] - prefix[i-1] [OK]

Common Mistakes:

• Subtracting prefix[i] instead of prefix[i-1]
• Adding prefix sums instead of subtracting
• Using wrong indices for prefix array