Maximum Sum Rectangle in 2D Matrix
A hard Dynamic Programming problem included in Love Babbar 450. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Dynamic Programming
- Sheets
- 1
- Core for
- 9 roles
- Platform
- GeeksforGeeks
The problem
Given a 2D matrix of integers, find the maximum sum of any rectangular submatrix within it.
Example 1
- Input
- matrix = [[1,2,-1,-4,-20],[-8,-3,4,2,1],[3,8,10,1,3],[-4,-1,1,7,-6]]
- Output
- 29
- Why
- The maximum sum rectangle is the submatrix from rows 1-2, columns 1-3, summing to 29.
Example 2
- Input
- matrix = [[-1,-2],[-3,-4]]
- Output
- -1
- Why
- The least negative element is -1, which is the maximum sum rectangle.
Example 3
- Input
- matrix = [[5]]
- Output
- 5
- Why
- Single element matrix, the maximum sum is that element.
Constraints
- 1 <= matrix.length, matrix[i].length <= 100
- -100 <= matrix[i][j] <= 100
How to think about it
Updated 2026-09-09Fix the top and bottom row boundaries of the rectangle. Compressing all rows between those boundaries into a 1D column-sum array turns the 2D problem into finding the maximum subarray sum via Kadane's algorithm.
Approaches, worst first
Brute force 4D submatrix evaluation
time O(R^2 * C^2) · space O(R * C)
Check all pairs of top-left and bottom-right corners, summing internal elements with 2D prefix sums. Requires checking O(R^2 * C^2) submatrices.
Row compression with Kadane's algorithmWrite this one
time O(R^2 * C) · space O(C)
Iterate top row r1 from 0 to R - 1. Maintain an array `colSums` of size C reset to 0. Iterate bottom row r2 from r1 to R - 1: add row r2 to colSums, then run 1D Kadane's algorithm on colSums in O(C) time.
Where people lose marks · 3
- All-negative matrix: initializing max sum to 0 causes Kadane's to return 0 instead of the least negative single cell value.
- Choosing rows vs columns for compression: if rows > cols, transpose or compress columns to keep the outer double-loop bounded by the smaller dimension.
- Forgetting to clear colSums array when advancing to a new top row r1.
The theory behind it
Dynamic Programming — the ground this problem stands on. All Dynamic Programming problems
What Dynamic Programming is
Dynamic programming is a method for solving a complex problem by breaking it into overlapping subproblems, solving each subproblem only once, and remembering the answers in a lookup table. Instead of recalculating identical questions over and over, future steps look up previous answers directly. By assembling these saved pieces from the bottom up or storing them during recursion, a task that would take billions of steps finishes in a fraction of a second.
When to reach for it
Reach for dynamic programming when questions ask for the maximum profit, minimum cost, total number of distinct ways to achieve a goal, or whether a target can be formed. Signals include overlapping choices where making a choice now affects what choices remain later, but greedy picking fails to find the true global optimum. If drawing a recursive decision tree reveals the same subproblem states repeating across branches, dynamic programming is needed.
How the pattern works
Identify the state variables that uniquely describe a subproblem, such as an array index and remaining capacity. Write the base cases first, representing states whose answers are known without calculation. Next, write the recurrence relation that expresses the current state using previously solved states, taking the minimum, maximum, or sum among your options. Build the solution either top-down by caching recursive returns in a memo table, or bottom-up by filling an array in topological dependency order. When each state depends only on the previous row, compress storage down to a single array.
What each operation costs
| Operation | Time |
|---|---|
| fill dynamic programming table of n states | O(n) |
| solve two-dimensional grid of m by n states | O(m * n) |
| space-optimized state transition keeping one row | O(n) |
What usually goes wrong with Dynamic Programming
- Filling a bottom-up table in an order where the current cell needs values that have not been computed yet, reading uninitialized zeros.
- Failing to initialize base cases properly, such as filling a minimization table with zeros instead of infinity, which traps the answer at zero.
- Overwriting values in a 1D space-optimized knapsack array by scanning in the wrong direction, allowing the same item to be chosen multiple times.
Which roles need this problem
Dynamic Programming is a core topic for these 9 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 5 more roles, including Game Developer, Cryptography Engineer, Performance Engineer.
Companies that have asked it
Tags taken from the problem's own GeeksforGeeks page — not a copied list.
Track this in your role's order
Pick your target role and all 370 problems — including this one — resequence to what that interview actually asks. Free.
Start freeMore Dynamic Programming problems
Problem set and role mapping as of .