If you’ve ever solved a problem that required adding up parts of an array or finding the sum of a subarray multiple times, you know how tedious it can get. What if there were a way to do it in a snap? Prefix sum is the algorithmic shortcut you didn’t know you needed—a simple yet powerful tool that turns repetitive tasks into lightning-fast solutions.
In this blog, we’ll introduce you to prefix sum, explain why it’s important, and show how it works in easy-to-understand terms. By the end, you’ll see why prefix sum is a game-changer for both coding interviews and real-world applications.
Understanding Why We Use Prefix Sum
Imagine you’re working with a large dataset, like a financial report or temperature readings over a year, and you need to:
• Calculate the total revenue for specific months.
• Find the average temperature over any given week.
• Compare data across different ranges.
Without prefix sum, you’d have to sum the elements in your range repeatedly—a time-consuming and inefficient process.
Enter Prefix Sum.
With a little bit of upfront work, prefix sum allows you to instantly calculate the sum of any range in an array. No loops. No wasted time. Just one simple subtraction.
What Is Prefix Sum?
The prefix sum array is created by keeping a running total of the elements in the original array. Each position in the prefix sum array represents the sum of all elements from the beginning of the array up to that index.
Let’s break it down with an example.
Given Array: [2, 4, 6, 8]
To create the prefix sum array:
1. Start with the first element: 2.
2. Add the second element: 2 + 4 = 6.
3. Add the third element: 6 + 6 = 12.
4. Add the fourth element: 12 + 8 = 20.
Prefix Sum Array: [2, 6, 12, 20]
Important: The prefix sum is usually stored in a separate array for flexibility and clarity, but in memory-constrained situations, you can modify the original array in place. The choice depends on the specific problem and constraints:
• Separate Array = Safer, more versatile.
• In-Place = Space-efficient but risky.
Subarray Sums And How They Correlate With Prefix Sum
Prefix sum is a powerful algorithmic pattern that simplifies the process of calculating subarray sums. However, its importance lies in its ability to efficiently address the broader problem: analyzing and working with subarray sums.
In real-world applications, analyzing segments of data is more important than looking at the whole array. Subarray sums allow us to focus on specific portions of data efficiently.
Now that we know why subarray sums are important, let’s connect the dots: prefix sum simplifies the process of calculating subarray sums, making previously tedious tasks lightning fast.
The Old Way: Brute Force
To find the sum of a subarray using brute force:
1. Start at index i.
2. Sum all elements up to index j.
3. Repeat for each possible subarray.
This approach has a time complexity of O(n^2), which is inefficient for large arrays.
The Prefix Sum Way
With prefix sum:
1. Precompute the prefix sum array in O(n).
2. Calculate the sum of any subarray using prefixSum[j] – prefixSum[i-1] in O(1).
Example:
Given nums = [2, 4, 6, 8], find the sum of the subarray from index 1 to 3:
1. Prefix sum array: [2, 6, 12, 20]
2. Subarray sum = prefixSum[3] – prefixSum[0] = 20 – 2 = 18
This efficiency is why prefix sum is so powerful.
Standard Approach: Using a Separate Array
Step 1: Create the Prefix Sum Array
In most cases, the prefix sum is stored in a new array.
Why?
• The original array might still be needed for other calculations or comparisons.
• Keeping a separate array ensures data integrity and makes debugging easier.
int[] prefixSum = new int[nums.length];
prefixSum[0] = nums[0];
for (int i = 1; i < nums.length; i++) {
prefixSum[i] = prefixSum[i - 1] + nums[i];
}• Here, prefixSum is a new array that stores cumulative sums.
• The original array nums remains unchanged.
Step 2: Formula for Subarray Sum
Once the prefix sum array is built, you can calculate the sum of any subarray using this formula:
sum = prefixSum[j] - prefixSum[i - 1];• Precomputing the prefix sum array takes O(n) time.
• Each subarray sum query is calculated in O(1) time.
• This makes prefix sums ideal for problems involving multiple subarray sum queries.
Real-World Example: Daily Expenses
Imagine you are tracking your daily expenses for a week, and you want to:
1. Find out how much you spent on specific days (a subarray of days).
2. Quickly calculate the total expenses for any given range of days.
We can use the prefix sum technique to achieve this efficiently.
Scenario
You have the following daily expenses:
• Day 1: $10
• Day 2: $20
• Day 3: $15
• Day 4: $30
• Day 5: $25
Represented as an array:
expenses = [10, 20, 15, 30, 25]
Now, let’s say you want to find the total expenses from:
• Day 2 to Day 4
public class PrefixSumExample {
public static void main(String[] args) {
int[] expenses = {10, 20, 15, 30, 25};
int[] prefixSum = new int[expenses.length + 1];
prefixSum[0] = 0;
for (int i = 1; i <= expenses.length; i++) {
prefixSum[i] = prefixSum[i - 1] + expenses[i - 1];
}
System.out.println("Prefix Sum Array: " + java.util.Arrays.toString(prefixSum));
int startDay = 2;
int endDay = 4;
int totalExpenses = prefixSum[endDay] - prefixSum[startDay - 1];
System.out.println("Total expenses from Day " + startDay + " to Day " + endDay + ": $" + totalExpenses);
}
}Conclusion
The prefix sum algorithm pattern is more than just a tool for solving subarray sum problems—it’s a gateway to efficient problem-solving in the world of arrays. By precomputing cumulative sums, prefix sum reduces repetitive calculations, saves time, and opens the door to elegant solutions for a wide range of problems, from simple range queries to complex optimization challenges. Whether you’re preparing for coding interviews or tackling real-world data analysis tasks, mastering prefix sum equips you with the ability to think ahead, optimize your approach, and solve problems with confidence. Start practicing today, and watch as prefix sum transforms the way you approach algorithmic challenges!
