Why Prefix Sum Array in DSA Javascript?

The Big Idea

Discover how a simple trick can turn slow, repeated adding into instant answers!

The Scenario

Imagine you have a long list of daily expenses and you want to quickly find out how much you spent between two specific 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 each query is slow and tiring, especially if you have many queries.

It's easy to make mistakes by missing a number or adding the wrong ones.

This wastes time and causes frustration.

The Solution

A prefix sum array keeps a running total of expenses up to each day.

This way, you can find the sum between any two days by subtracting two numbers, making queries super fast and easy.

Before vs After

✗ Before

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

✓ After

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

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

What It Enables

It enables lightning-fast queries for sums over any range, even in huge lists.

Real Life Example

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

Key Takeaways

Manual sum queries are slow and error-prone.

Prefix sum arrays store running totals for quick subtraction.

This makes range sum queries fast and reliable.

Practice

1. What does a prefix sum array represent for a given array arr?

easy

A. An array where each element is the sum of all elements from the start up to that index in arr.

B. An array where each element is the product of all elements from the start up to that index in arr.

C. An array where each element is the difference between consecutive elements in arr.

D. An array where each element is the maximum value found in arr up to that index.

Solution

1. Step 1: Understand prefix sum array definition

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

2. Step 2: Compare options with definition

   Only An array where each element is the sum of all elements from the start up to that index in arr. correctly describes this sum accumulation behavior.

3. Final Answer:

   An array where each element is the sum of all elements from the start up to that index in arr. -> Option A

4. Quick Check:

   Prefix sum = cumulative sum [OK]

Hint: Prefix sum means cumulative addition from start [OK]

Common Mistakes:

• Confusing prefix sum with product or max
• Thinking prefix sum stores differences
• Assuming prefix sum stores original values

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

easy

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

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

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

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

Solution

1. Step 1: Check initialization of prefix sum array

   Prefix sum starts with first element of arr, so prefix[0] = arr[0].

2. Step 2: Verify loop logic for prefix sums

   Each prefix[i] = prefix[i-1] + arr[i] correctly accumulates sums.

3. Final Answer:

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

4. Quick Check:

   Initialize with arr[0], then add previous prefix [OK]

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

Common Mistakes:

• Starting prefix with 0 instead of arr[0]
• Using multiplication instead of addition
• Accessing prefix[-1] which is undefined

3. What will be the output of the following JavaScript code?

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

medium

A. [2, 6, 10, 18]

B. [2, 4, 6, 8]

C. [2, 8, 14, 22]

D. [2, 6, 12, 20]

Solution

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

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

2. Step 2: Confirm final prefix array

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

3. Final Answer:

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

4. Quick Check:

   Sum accumulates correctly [OK]

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

Common Mistakes:

• Adding elements incorrectly or skipping first element
• Confusing prefix array with original array
• Off-by-one errors in loop

4. Identify the error in the following code that attempts to create a prefix sum array:

const arr = [1, 3, 5, 7];
const prefix = [];
for(let i = 0; i < arr.length; i++) {
  prefix[i] = prefix[i-1] + arr[i];
}
console.log(prefix);

medium

A. prefix array is not initialized with first element, causing prefix[-1] to be undefined.

B. The loop should start from i = 1 instead of 0.

C. The code uses addition instead of multiplication.

D. The array arr is empty, so prefix will be empty.

Solution

1. Step 1: Analyze prefix[-1] usage at i=0

   At i=0, prefix[i-1] is prefix[-1], which is undefined, causing NaN result.

2. Step 2: Fix initialization to avoid undefined

   Initialize prefix[0] = arr[0] before loop or handle i=0 separately.

3. Final Answer:

   prefix array is not initialized with first element, causing prefix[-1] to be undefined. -> Option A

4. Quick Check:

   Accessing prefix[-1] causes undefined [OK]

Hint: Initialize prefix[0] before loop to avoid undefined [OK]

Common Mistakes:

• Starting loop at 0 without initializing prefix[0]
• Ignoring undefined prefix[-1] access
• Assuming prefix[-1] is zero automatically

5. Given an array arr = [3, 1, 4, 1, 5], how can you use a prefix sum array to efficiently find the sum of elements between indices 1 and 3 (inclusive)?

hard

A. Calculate prefix sums, then subtract prefix[0] from prefix[4].

B. Calculate prefix sums, then subtract prefix[0] from prefix[3].

C. Calculate prefix sums, then subtract prefix[0] from prefix[2].

D. Calculate prefix sums, then subtract prefix[1] from prefix[3].

Solution

1. Step 1: Build prefix sum array for arr

   prefix = [3, 4, 8, 9, 14] where prefix[i] = sum of arr[0..i].

2. Step 2: Calculate sum from index 1 to 3

   Sum = prefix[3] - prefix[0] = 9 - 3 = 6 (elements 1,4,1).

3. Final Answer:

   Calculate prefix sums, then subtract prefix[0] from prefix[3]. -> Option B

4. Quick Check:

   Sum(1 to 3) = prefix[3] - prefix[0] [OK]

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

Common Mistakes:

• Subtracting prefix[i] instead of prefix[i-1]
• Using wrong indices for subtraction
• Not building prefix sums first