The “Minimum Falling Path Sum” problem asks you to find the smallest possible sum of a falling path through a square matrix. A falling path starts at any cell in the first row and moves downward one row at a time until it reaches the last row. From a cell, you can move straight down, down-left, or down-right, as long as the destination is within bounds. The goal is to minimize the total sum of the visited cells.
This problem looks like a grid traversal task, but it is really an optimization problem with overlapping choices. Each step offers up to three options, and the choice you make now affects what options are available later. That structure makes it a strong candidate for dynamic programming.
Why greedy choices fail
A tempting strategy is to pick the smallest number in the top row and then keep choosing the smallest available neighbor as you move downward. This can fail because a locally small value may funnel you into a region of the matrix with high costs later, while a slightly wider early choice could lead to a much cheaper overall path.
The key issue is that you cannot judge the quality of a move based only on its immediate cost. You need to consider the best possible future continuation from that cell. This dependence on future outcomes is exactly what dynamic programming is built to handle.
Want to explore more coding problem solutions? Check out the Minimum Cost for Tickets and Reconstruct Itinerary.
Recognizing the optimal substructure
A crucial insight is that the best falling path that ends at a particular cell in row r depends only on the best falling paths that end at certain cells in row r-1. Specifically, to reach a cell, you must have come from one of the three allowed positions above it.
This means you can define a subproblem: the minimum path sum to reach each cell. Once you know these values for one row, you can compute them for the next row. This creates a clean bottom-up progression through the matrix.
Dynamic programming state: minimum cost to reach each cell
The most natural state definition is the minimum falling path sum to reach each cell at a given row. For the first row, the minimum cost to reach each cell is simply its own value, because the path starts there.
For every subsequent row, the minimum cost to reach a cell is its own value plus the minimum of the reachable costs from the row above. This transition captures the allowed moves and ensures that each cell’s cost accounts for the best possible path leading into it.
Handling boundaries without special tricks
Boundary cells on the left and right edges have fewer incoming options because down-left or down-right may fall outside the matrix. A good solution handles these cases naturally by only considering valid predecessor cells.
This keeps the transition rule consistent and avoids fragile logic. The algorithm remains the same everywhere; only the set of valid predecessors changes near edges.
Why this approach guarantees the minimum total sum
By computing the minimum cost to reach every cell in the last row, you ensure that every possible falling path is accounted for. Each path ends at some cell in the bottom row, and the dynamic programming table contains the minimum cost to reach each of those endpoints.
The final answer is therefore the minimum value among the computed costs in the last row. This aligns directly with the problem definition and guarantees correctness.
Space optimization: reusing rows
Although the most straightforward dynamic programming implementation stores a full matrix of computed costs, you do not actually need all rows at once. Each row depends only on the row directly above it.
This allows you to reuse space by keeping only the previous row’s costs while computing the current row. This reduces memory usage significantly while preserving the same correctness and runtime behavior.
Time and space complexity considerations
The algorithm processes each cell once and performs a constant amount of work per cell, making the runtime proportional to the number of cells in the matrix. Space usage depends on whether you store a full table or optimize to a single rolling row, but both are efficient for typical constraints.
This balance is one reason the problem is a classic introduction to dynamic programming on grids.
Why this problem is common in interviews
Minimum Falling Path Sum appears frequently in interviews because it tests fundamental dynamic programming intuition. It evaluates whether you can identify optimal substructure, define the right state, and build a correct transition rule.
It also checks whether you can handle boundary conditions cleanly without complicating the logic.
What this problem teaches beyond falling paths
Beyond this specific task, the problem teaches a reusable pattern: when a grid path problem asks for a minimum or maximum total under constrained moves, dynamic programming is often the correct approach. Similar structures appear in seam carving, path cost optimization, and many scheduling-like grid problems.
If you can clearly explain why greedy fails, how the minimum-to-reach state is defined, and why the final answer comes from the last row’s minimum, you demonstrate strong algorithmic reasoning. That depth of understanding makes “Minimum Falling Path Sum” an excellent exercise in grid-based dynamic programming.
Top comments (0)