Why Prefix Sum Array in DSA Typescript?

The Big Idea

What if you could answer any sum question instantly without adding numbers again and again?

The Scenario

Imagine you have a long list of daily expenses and you want to quickly find out how much you spent between any two days. Doing this by adding each day's expense every time you ask is like counting coins one by one every time you want to know your total.

The Problem

Manually adding numbers for every query is slow and tiring. If you have many queries, you waste a lot of time repeating the same additions. It's easy to make mistakes, especially with long lists, and it feels like doing the same work again and again.

The Solution

A prefix sum array keeps a running total of expenses up to each day. This way, to find the total between two days, you just subtract two numbers. It's like having a ready-made summary that saves you from counting coins repeatedly.

Before vs After

✗ Before

function sumRange(expenses: number[], start: number, end: number): number {
  let total = 0;
  for (let i = start; i <= end; i++) {
    total += expenses[i];
  }
  return total;
}

✓ After

function createPrefixSum(expenses: number[]): number[] {
  const prefixSum = [0];
  for (let i = 0; i < expenses.length; i++) {
    prefixSum.push(prefixSum[i] + expenses[i]);
  }
  return prefixSum;
}

function sumRange(prefixSum: number[], start: number, end: number): number {
  return prefixSum[end + 1] - prefixSum[start];
}

What It Enables

It makes answering sum queries super fast and easy, even for huge lists.

Real Life Example

Bank apps use prefix sums to quickly show your spending between any two dates without recalculating every time you check.

Key Takeaways

Manually summing ranges is slow and error-prone.

Prefix sum arrays store running totals to speed up queries.

They make repeated sum queries efficient and simple.

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