🚀 Optimize Array Problems | Crack Range Queries | Win Interviews

🔥 Why Prefix Sum Is a Game Changer

Prefix Sum is one of the most powerful optimization techniques in Data Structures and Algorithms.

It helps you convert

slow repeated calculations

into fast constant-time answer

If you struggle with array problems where sums are calculated again and again, Prefix Sum is the missing piece.

This concept appears frequently in

coding interviews

competitive programming

dynamic programming foundations

👨‍🎓 Who Should Learn Prefix Sum

This article is perfect if

✅ You already know array traversal

✅ You want to move from brute force to optimized solutions

✅ You are preparing for product-based company interviews

✅ You want clarity instead of memorizing tricks

🧠 What Exactly Is Prefix Sum?

Prefix Sum means precomputing cumulative sums of an array.

In simple terms

Each index stores the sum of all elements before it and including it.

Example

Original Array: [2, 4, 6, 8]

Prefix Sum: [2, 6, 12, 20]

So

index 0 → sum till 0

index 1 → sum till 1

index 2 → sum till 2

🌍 Real-Life Analogy (Very Important)

Think about your bank balance.

Today’s balance = Yesterday’s balance + Today’s transaction

You never recalculate your entire account history every day.

That running balance is exactly how Prefix Sum works.

Other relatable examples

monthly expense tracking

cricket scoreboard totals

rainfall accumulation over days

⚙️ How Prefix Sum Works (Core Logic)

Let’s take this array

arr = [3, 1, 4, 2]

We build a new array step by step.

prefix[0] = 3
prefix[1] = 3 + 1 = 4
prefix[2] = 4 + 4 = 8
prefix[3] = 8 + 2 = 10

🔑 Universal Formula

prefix[i] = prefix[i - 1] + arr[i]

This formula is the heart of prefix sum.

🧪 Step-by-Step Dry Run (Interview Favorite)

Given

arr = [5, 10, 15]

Execution flow

prefix[0] = 5
prefix[1] = 5 + 10 = 15
prefix[2] = 15 + 15 = 30

Final prefix array

[5, 15, 30]

Once built, this array answers sum queries instantly.

💻 JavaScript Implementation

🛠️ Build Prefix Sum Array

const arr = [5, 10, 15];
const prefix = [];
prefix[0] = arr[0];
for (let i = 1; i < arr.length; i++) {
  prefix[i] = prefix[i - 1] + arr[i];
}
console.log(prefix); // [5, 15, 30]

Clean, readable, interview-ready.

⚡ Range Sum Query Using Prefix Sum

❓ Problem

Find the sum of elements between index L and R (inclusive).

Example input

arr = [2, 4, 6, 8, 10]
L = 1
R = 3

🐌 Brute Force Approach

4 + 6 + 8 = 18
Time complexity per query is O(n)

This becomes slow when queries increase.

🚀 Optimized Using Prefix Sum

Prefix array

[2, 6, 12, 20, 30]

Formula used

sum(L, R) = prefix[R] - prefix[L - 1]

Dry run

sum(1, 3) = 20 - 2 = 18
Each query now runs in O(1) time.

💻 JavaScript Code for Range Query

function rangeSum(prefix, L, R) {
  if (L === 0) {
    return prefix[R];
  }
  return prefix[R] - prefix[L - 1];
}

⏱️ Time and Space Complexity

Building prefix sum takes linear time.

Each query executes in constant time.

Extra memory used is linear.

This is a classic time–space trade-off.

🎯 Interview Problems Where Prefix Sum Is Used

Range sum queries

Subarray sum equals K

Equilibrium index

Maximum subarray sum (Kadane’s foundation)

Difference array problems

If you see repeated sums, prefix sum is usually the answer.

⚠️ Common Mistakes to Avoid

> Forgetting the special case when L equals zero

> Recomputing sums instead of using prefix array

> Overwriting the original array

> Confusing prefix sum with sliding window

Interviewers often check these edge cases.

🆚 Prefix Sum vs Sliding Window

Prefix Sum works best when

the array is static

multiple range queries are asked

Sliding Window works best when

the window moves

space needs to be optimized

In real interviews, both are often combined.

🔗 What to Read Next (Internal Linking)
👉 Data Structures and Algorithms – Complete Roadmap (2026)

👉 Array Traversal – Beginner to Interview Level

👉 Sliding Window Technique Explained Step by Step

👉 Kadane’s Algorithm Explained Simply

Previous Topic -

https://www.dsawithpiyush.com/post/array-traversal-in-data-structure-or-ds

🏆 Final Advice from DSA With Piyush

If a problem feels slow, look for repeated work.

If work repeats, Prefix Sum is your first optimization tool