Prefix Sum: Beginner

Published: 3 days ago (February 24, 2026 at 09:18 PM EST)

5 min read

Source: Dev.to

If you’ve ever looked at a coding challenge and felt like you were staring at a brick wall, you aren’t alone. Most of the time, the secret isn’t about being a math genius—it’s about recognising the pattern.

In this series we break down the most essential coding patterns—from beginner to expert level—keeping things simple, practical, and (hopefully) a little bit fun.

Today we’re diving deep into Prefix Sum.

Whether you’re preparing for a big interview or just trying to write cleaner, faster code, mastering this pattern is a total game‑changer.

Let’s strip away the jargon and see how it actually works!

What is Prefix Sum?

Imagine you’re standing at the start of a long road, and every mile there is a sign telling you how many coins are buried under the ground at that specific spot.

If I asked, “How many coins are there between mile 3 and mile 8?”, you’d normally have to walk from 3 to 8 and add them up one by one.

That works for a short walk, but what if the road is 10 000 miles long and I ask you 1 000 different questions? You’d be exhausted!

Prefix Sum is a preprocessing technique that creates a “running total” of an array. By doing a little work upfront, you can answer any “range‑sum” query in O(1) time—turning a repetitive addition task into a simple subtraction.

The Mathematical Logic

We create a new array (let’s call it prefix) where each element at index i stores the sum of all elements from the start of the original array arr up to that index.

Formula

prefix[i] = arr[0] + arr[1] + … + arr[i]

More Efficient Recurrence

prefix[i] = prefix[i‑1] + arr[i]   (for i > 0)

A Simple Example

arr = [3, 1, 4, 1, 5]

Index 0: 3
Index 1: 3 + 1 = 4
Index 2: 4 + 4 = 8
Index 3: 8 + 1 = 9
Index 4: 9 + 5 = 14

prefix = [3, 4, 8, 9, 14]

To get the sum of elements from index 2 to 4 (4, 1, 5):

result = prefix[4] – prefix[1] = 14 – 4 = 10

Why this works:

prefix[4] = A0 + A1 + A2 + A3 + A4
prefix[1] = A0 + A1
=> prefix[4] – prefix[1] = A2 + A3 + A4

Efficiency and Complexity

Time Complexity

Without Prefix Sum: each range query costs O(N), so M queries cost O(M × N).
With Prefix Sum: building the prefix array costs O(N) once; each query is O(1).

Space Complexity

We usually allocate an extra array of size N, giving O(N) auxiliary space.
If we overwrite the original array, we can achieve O(1) extra space.

If you need a quick refresher on Big‑O notation, check out my previous guide here.

Where Is It Used? (Applications)

Financial & Sales Analytics – Instant totals for any custom date range.
Image Processing – 2‑D Prefix Sum (Summed‑Area Table) enables constant‑time rectangle sums for blurring, feature detection, etc.
Web Analytics – Cumulative visitors or active users over time‑series data.
Database Query Optimization – Speed up range scans and aggregate queries.
Game Development – Cumulative scores, resources, or 2‑D lighting/shadow calculations.

Classic Interview Problems

Range Sum Query – Repeatedly find the sum between two indices.
Pivot Index – Find an index where the sum of elements to the left equals the sum to the right.
Subarray Sum Equals K – Combine Prefix Sum with a hash map to count subarrays that sum to K.
Maximum Subarray (Kadane’s variant) – Use Prefix Sums to compute prefix minima/maxima.

Recognising these patterns early can save you a ton of time and stress in interviews.

Implementation

Now, let’s see this pattern in action!

For these demonstrations, I’ll be using C#. However, don’t let the language choice stop you if you usually code in Python, Java, JavaScript, etc. The logic and the underlying pattern remain exactly the same across all languages—only the syntax changes. Feel free to follow along and implement the logic in your preferred language.

using System;

public class PrefixSumExample
{
    public static void Main()
    {
        int[] nums = { 3, 1, 4, 1, 5 };

        // Preprocessing: Build the Prefix Sum Array
        int[] prefixSum = BuildPrefixSum(nums);

        // Querying: Get sum of range [index 1 to 3] (values: 1, 4, 1)
        // Expected: 6
        int result = GetRangeSum(prefixSum, 1, 3);

        Console.WriteLine("\nSum of range [1 to 3]: " + result);
    }

    public static int[] BuildPrefixSum(int[] A)
    {
        int n = A.Length;
        int[] P = new int[n];

        // The first element is always the same
        P[0] = A[0];

        for (int i = 1; i  // Incomplete loop shown as in the original source
    }
}

Don’t get discouraged if you get stuck! The goal isn’t to solve each problem in five minutes; the goal is to recognise the pattern. Happy coding!

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