Minimum Falling Path Sum
A medium Dynamic Programming interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Dynamic Programming
- Sheets
- 1
- Core for
- 9 roles
The problem
In a square grid, pick one cell in the top row and move down one row at a time until you reach the bottom. From each cell you may go to the cell directly below or to either of its diagonal neighbours below. Return the smallest sum of cells such a descent can have.
Example 1
- Input
- matrix = [[5,2,6],[3,7,1],[4,8,9]]
- Output
- 9
- Why
- The cheapest fall is 2, then 3, then 4: total 9.
Example 2
- Input
- matrix = [[-10,20],[-30,-5]]
- Output
- -40
- Why
- Minimum falling path: -10 -> -30, sum = -40.
Example 3
- Input
- matrix = [[-48]]
- Output
- -48
- Why
- Single cell grid, the minimum falling path sum is the cell value.
Constraints
- n == matrix.length == matrix[i].length
- 1 <= n <= 100
- -100 <= matrix[i][j] <= 100
How to think about it
Updated 2026-09-29At each cell (r, c), a falling path can only arrive from three potential predecessors on row r - 1: directly above (r-1, c), diagonally left (r-1, c-1), and diagonally right (r-1, c+1). Taking the minimum of the available predecessors and adding matrix[r][c] propagates the best sum forward.
Approaches, worst first
Recursive path tree
time O(3^n) · space O(n)
From every cell in the first row, branch into all three downward choices. Explores 3^n branches, recalculating overlapping suffix paths.
2D dynamic programming table
time O(n^2) · space O(n^2)
dp[r][c] = matrix[r][c] + min(dp[r-1][c-1], dp[r-1][c], dp[r-1][c+1]), guarding boundary columns c = 0 and c = n - 1. Return the minimum element across the final row.
Two-row rolling arrayWrite this one
time O(n^2) · space O(n)
Maintain only the previous row's minimal path sums and the current row's buffer. Updates run in-place or across two alternating rows of size n.
Where people lose marks · 3
- Out-of-bounds column lookups: c - 1 is invalid when c = 0, and c + 1 is invalid when c = n - 1. Boundary checks must ignore non-existent diagonal parents rather than reading uninitialized values.
- Negative cell values: initializing DP minimum accumulators with 0 or positive infinity defaults distorts paths through negative numbers.
- Single-element matrix (1x1): the minimum falling path is just matrix[0][0].
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.
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