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 and use a prefix sum array.

How does the time needed grow as the input array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

void buildPrefixSum(int arr[], int prefix[], int n) {
    prefix[0] = arr[0];
    for (int i = 1; i < n; i++) {
        prefix[i] = prefix[i - 1] + arr[i];
    }
}

int rangeSum(int prefix[], int l, int r) {
    if (l == 0) return prefix[r];
    return prefix[r] - prefix[l - 1];
}
    

This code builds a prefix sum array from an input array and then uses it to find the sum of elements between two indexes quickly.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The for-loop that runs from 1 to n-1 to build the prefix sums.
• How many times: It runs exactly n-1 times, once for each element after the first.

How Execution Grows With Input

As the input array size grows, the number of steps to build the prefix sum grows roughly the same amount.

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

Pattern observation: The work grows linearly with input size; doubling the input roughly doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to build the prefix sum array grows in direct proportion to the size of the input array.

Common Mistake

[X] Wrong: "Calculating the sum of any range using the prefix sum array takes O(n) time because we still add elements."

[OK] Correct: Using the prefix sum array, range sums are found with just two lookups and one subtraction, which is O(1) time, not O(n).

Interview Connect

Understanding prefix sums helps you quickly answer range sum queries, a common pattern in coding problems. Mastering this shows you can optimize repeated calculations efficiently.

Self-Check

"What if we needed to update elements in the array frequently? How would that affect the time complexity of using a prefix sum array?"

Practice

1. What is the main purpose of a prefix sum array in data structures?

easy

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

B. To sort the array elements in ascending order

C. To store the product of all elements

D. To reverse the array efficiently

Solution

1. Step 1: Understand prefix sum array definition

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

2. Step 2: Identify its use

   This allows quick calculation of sums for any subarray by subtracting prefix sums.

3. Final Answer:

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

4. Quick Check:

   Prefix sum = fast subarray sums [OK]

Hint: Prefix sums help find subarray sums fast [OK]

Common Mistakes:

• Confusing prefix sum with sorting
• Thinking it stores products
• Assuming it reverses arrays

2. Which of the following is the correct way to initialize a prefix sum array prefix for an integer array arr of size n in C?

easy

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

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

C. for(int i=0; i

D. prefix[0] = 0; for(int i=1; i<=n; i++) prefix[i] = prefix[i-1] + arr[i];

Solution

1. Step 1: Initialize first prefix element

   Set prefix[0] to arr[0] to start sums correctly.

2. Step 2: Loop from 1 to n-1 adding previous prefix and current arr

   Use prefix[i] = prefix[i-1] + arr[i] for correct cumulative sums.

3. Final Answer:

   prefix[0] = arr[0]; for(int i=1; i -> Option B

4. Quick Check:

   Start at arr[0], add cumulatively [OK]

Hint: Start prefix with arr[0], then add cumulatively [OK]

Common Mistakes:

• Starting prefix at 0 and indexing arr out of bounds
• Copying arr directly without summing
• Using wrong starting index for arr

3. Given the array arr = {2, 4, 6, 8}, what is the prefix sum array after running the code:

int prefix[4];
prefix[0] = arr[0];
for(int i = 1; i < 4; i++) {
    prefix[i] = prefix[i-1] + arr[i];
}

medium

A. [0, 2, 6, 12]

B. [2, 4, 6, 8]

C. [2, 6, 12, 20]

D. [2, 8, 14, 22]

Solution

1. Step 1: Calculate prefix[0]

   prefix[0] = arr[0] = 2

2. Step 2: Calculate prefix sums for i=1 to 3

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

3. Final Answer:

   [2, 6, 12, 20] -> Option C

4. Quick Check:

   Sum accumulates: 2, 6, 12, 20 [OK]

Hint: Add current element to previous prefix sum [OK]

Common Mistakes:

• Confusing prefix sums with original array
• Starting sums from zero incorrectly
• Adding wrong elements in loop

4. Identify the error in the following code snippet that attempts to create a prefix sum array for arr of size n:

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

medium

A. prefix[0] should be initialized to arr[0], not 0

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

C. Array prefix should be of size n+1

D. The code will cause a runtime error

Solution

1. Step 1: Check prefix[0] initialization

   prefix[0] must hold arr[0] to start sums correctly, not zero.

2. Step 2: Analyze loop correctness

   Loop starting at 1 is correct if prefix[0] is set properly.

3. Final Answer:

   prefix[0] should be initialized to arr[0], not 0 -> Option A

4. Quick Check:

   prefix[0] = arr[0] starts sums right [OK]

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

Common Mistakes:

• Setting prefix[0] to zero causes wrong sums
• Starting loop at wrong index
• Assuming prefix array size must be n+1

5. You have an array arr = {3, 1, 4, 1, 5, 9, 2}. Using a prefix sum array, how would you efficiently find the sum of elements from index 2 to 5 (inclusive)?

hard

A. Calculate prefix[5] + prefix[2]

B. Sum elements from arr[2] to arr[5] directly every time

C. Use prefix[6] - prefix[2] to get the sum

D. Calculate prefix[5] - prefix[1] after building prefix sums

Solution

1. Step 1: Build prefix sum array

   prefix[i] = sum of arr[0] to arr[i]. For example, prefix[5] = sum of arr[0..5].

2. Step 2: Calculate sum from index 2 to 5

   Sum = prefix[5] - prefix[1], because prefix[1] = sum of arr[0..1].

3. Final Answer:

   Calculate prefix[5] - prefix[1] after building prefix sums -> Option D

4. Quick Check:

   Sum(i..j) = prefix[j] - prefix[i-1] [OK]

Hint: Sum subarray = prefix[end] - prefix[start-1] [OK]

Common Mistakes:

• Adding prefix sums instead of subtracting
• Using wrong indices for prefix subtraction
• Summing elements directly ignoring prefix sums