Prefix Sum Array in DSA Typescript - 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.

function prefixSum(arr: number[]): number[] {
  if (arr.length === 0) return [];
  const prefix: number[] = [];
  prefix[0] = arr[0];
  for (let i = 1; i < arr.length; i++) {
    prefix[i] = prefix[i - 1] + arr[i];
  }
  return prefix;
}
    

This code creates a prefix sum array where prefix[i] is the sum of all elements from the start of the array up to and including arr[i].

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: A single loop that adds elements cumulatively.
• How many times: The loop runs once for each element after the first, so (n - 1) times.

How Execution Grows With Input

As the input array 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 number of operations grows directly with the input size.

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: "Since we add elements multiple times, the time is quadratic O(n²)."

[OK] Correct: Each element is added only once in the loop, so the total work grows linearly, not squared.

Interview Connect

Understanding prefix sums helps you quickly answer range sum queries, a common problem in coding challenges.

Self-Check

"What if we used recursion instead of a loop to build the prefix sum? 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 between any two indices

B. To sort the array elements in ascending order

C. To reverse the array elements

D. To find the maximum element in the array

Solution

1. Step 1: Understand what prefix sum array stores

   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 between any two indices -> Option A

4. Quick Check:

   Prefix sum array = fast range sum queries [OK]

Hint: Prefix sums help answer range sum queries fast [OK]

Common Mistakes:

• Confusing prefix sum with sorting
• Thinking prefix sum reverses array
• Assuming prefix sum finds max element

2. Which of the following TypeScript code snippets correctly initializes a prefix sum array prefix for an input array arr?

easy

A. const prefix = []; for(let i=0; i

B. const prefix = [0]; for(let i=0; i

C. const prefix = [arr[0]]; for(let i=1; i

D. const prefix = [0]; for(let i=1; i<=arr.length; i++) { prefix[i] = prefix[i-1] + arr[i]; }

Solution

1. Step 1: Check initialization and loop indexing

   const prefix = [0]; for(let i=0; i

2. Step 2: Identify errors in other options

   const prefix = []; for(let i=0; i

3. Final Answer:

   const prefix = [0]; for(let i=0; i -> Option B

4. Quick Check:

   Prefix sum array starts with 0 and accumulates sums [OK]

Hint: Start prefix with 0 and add arr[i] in loop [OK]

Common Mistakes:

• Accessing array out of bounds
• Not initializing prefix[0] to 0
• Using negative indices

3. Given the TypeScript code below, what is the output printed?

const arr = [2, 4, 6, 8];
const prefix = [0];
for(let i = 0; i < arr.length; i++) {
  prefix[i + 1] = prefix[i] + arr[i];
}
console.log(prefix);

medium

A. [2, 6, 12, 20]

B. [0, 2, 4, 6, 8]

C. [0, 2, 6, 12, 20]

D. [2, 4, 6, 8, 0]

Solution

1. Step 1: Calculate prefix sums step-by-step

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

2. Step 2: Confirm final prefix array

   The prefix array is [0, 2, 6, 12, 20].

3. Final Answer:

   [0, 2, 6, 12, 20] -> Option C

4. Quick Check:

   Sum accumulates correctly in prefix array [OK]

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

Common Mistakes:

• Forgetting initial 0 in prefix
• Adding wrong elements in loop
• Off-by-one errors in indexing

4. The following TypeScript code attempts to create a prefix sum array but has a bug. What is the error?

const arr = [1, 3, 5];
const prefix = [];
for(let i = 0; i < arr.length; i++) {
  prefix[i + 1] = prefix[i] + arr[i];
}
console.log(prefix);

medium

A. prefix array is not initialized with 0 at index 0

B. Loop runs one time too many

C. arr[i] is accessed out of bounds

D. prefix array is declared with wrong type

Solution

1. Step 1: Check prefix array initialization

   The prefix array is empty initially, so prefix[0] is undefined.

2. Step 2: Identify impact on prefix sum calculation

   prefix[i] is undefined for i=0, so prefix[1] = undefined + arr[0] results in NaN.

3. Final Answer:

   prefix array is not initialized with 0 at index 0 -> Option A

4. Quick Check:

   Initialize prefix[0] = 0 to avoid NaN [OK]

Hint: Always start prefix array with 0 at index 0 [OK]

Common Mistakes:

• Not initializing prefix[0]
• Assuming empty array has prefix[0]
• Ignoring NaN results

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 sums, then sum = prefix[5] - prefix[1]

B. Calculate prefix sums, then sum = prefix[6] - prefix[1]

C. Calculate prefix sums, then sum = prefix[5] - prefix[2]

D. Calculate prefix sums, then sum = prefix[6] - prefix[2]

Solution

1. Step 1: Understand prefix sum indexing

   Prefix sum array prefix has prefix[0] = 0, prefix[i] = sum of arr[0] to arr[i-1].

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

   Sum = prefix[6] - prefix[2], because prefix[6] sums arr[0..5], prefix[2] sums arr[0..1].

3. Final Answer:

   Calculate prefix sums, then sum = prefix[6] - prefix[2] -> Option D

4. Quick Check:

   Sum of arr[2..5] = prefix[6] - prefix[2] [OK]

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

Common Mistakes:

• Off-by-one errors in prefix indices
• Subtracting wrong prefix sums
• Confusing inclusive/exclusive ranges