Prefix Sums Explained Visually | Range Sum Queries in O(1)
Prefix sums are one of those ideas that look simple, but unlock a huge number of fast solutions.
In this video, I explain prefix sums visually from first principles:
how to build the prefix array in one pass
how to answer range sum queries in O(1)
why the subtraction formula works
how 2D prefix sums extend the same idea to grids
how difference arrays act like the inverse pattern for fast range updates
where this pattern shows up in interview and competitive programming problems
If you’ve seen range queries, subarray problems, or batch updates and felt like there must be a cleaner way, this is the mental model you want.
If you liked this video and explanation with visualization, drop a like, subscribe, and comment what you’d like to learn next.
Practice resources are below if you want to sharpen this concept with a structured set of problems.
#PrefixSums #DSA #Algorithms #CompetitiveProgramming #codinginterview
Want to practice prefix sums properly instead of solving random problems?
Here’s a structured path using trustworthy sources I recommend: CSES, LeetCode, cp-algorithms, and the Competitive Programmer’s Handbook.
Start with the core idea
1. Learn the basic range-sum pattern:
CSES Static Range Sum Queries
https://cses.fi/problemset/task/1646/
LeetCode 303. Range Sum Query - Immutable
https://leetcode.com/problems/range-s...
Then move to “recognizing prefix sums in disguise”
2. Practice subarray / hashmap variants:
LeetCode 560. Subarray Sum Equals K
https://leetcode.com/problems/subarra...
LeetCode 974. Subarray Sums Divisible by K
https://leetcode.com/problems/subarra...
LeetCode 523. Continuous Subarray Sum
https://leetcode.com/problems/continu...
Then extend the idea to 2D
3. Practice matrix / rectangle query variants:
LeetCode 304. Range Sum Query 2D - Immutable
https://leetcode.com/problems/range-s...
CSES Forest Queries
https://cses.fi/problemset/task/1652/
Then learn the inverse pattern: difference arrays
4. Practice range updates:
CSES Range Update Queries
https://cses.fi/problemset/task/1651/
LeetCode 2381. Shifting Letters II
https://leetcode.com/problems/shiftin...
LeetCode 1094. Car Pooling
https://leetcode.com/problems/car-poo...
LeetCode 1109. Corporate Flight Bookings
https://leetcode.com/problems/corpora...
Suggested order
Day 1: CSES Static Range Sum Queries, LeetCode 303
Day 2: LeetCode 560
Day 3: LeetCode 974, LeetCode 523
Day 4: LeetCode 304, CSES Forest Queries
Day 5: CSES Range Update Queries, LeetCode 2381
Day 6: LeetCode 1094, LeetCode 1109
What you should be able to spot after this set
“many range sum queries on a static array”
“count / detect subarrays using prefix sums + hashmap”
“rectangle sum queries on a grid”
“many range updates, then reconstruct final values”
Reference material
Competitive Programmer’s Handbook:
https://www.cses.fi/book/book.pdf
cp-algorithms Fenwick Tree article
useful for understanding when plain prefix sums stop being enough and when updates require a stronger structure:
https://cp-algorithms.com/data_struct...