LeetCode – Minimum Falling Path Sum with Memoization

What will you learn?

Explore the efficient solution to the Minimum Falling Path Sum problem on LeetCode using the memoization technique. Enhance your understanding of dynamic programming and optimization strategies.

Introduction to the Problem and Solution

In this challenge, you are presented with a square grid of integers representing an n x n matrix. The goal is to determine the minimum sum of a falling path through the grid. A falling path begins at any element in the first row and selects one element from each subsequent row. The chosen element in each subsequent row must be in a column that differs from the previous row’s column by at most one.

To tackle this problem effectively, we leverage dynamic programming combined with memoization. By storing previously computed results, we can eliminate redundant calculations and enhance our solution’s efficiency.


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def minFallingPathSum(matrix):
    n = len(matrix)
    memo = [[None] * n for _ in range(n)]

    def helper(row, col):
        if row == n:
            return 0

        if memo[row][col]:
            return memo[row][col]

        res = matrix[row][col] + min(
            helper(row + 1, new_col)
            for new_col in range(max(0, col - 1), min(n, col + 2))

        memo[row][col] = res
        return res

    return min(helper(0, col) for col in range(n))

# Copyright PHD


  • Initialize a 2D memo list to store computed results.
  • Define a recursive helper function to calculate the minimum falling path sum from each cell.
  • Utilize dynamic programming and memoization to optimize computations by avoiding redundancy.
  • Compute and return the minimum falling path sum by iterating over all columns in the first row.
    How does memoization improve performance?

    Memoization enhances performance by storing intermediate results and reusing them when necessary, thereby reducing time complexity significantly.

    Can I use tabulation instead of recursion with memoization?

    Yes, tabulation (bottom-up approach) can also be employed instead of recursion with memoization for effective dynamic programming problem-solving.

    Is it necessary to create a separate helper function for recursion?

    While not mandatory, creating a distinct helper function often leads to cleaner code structure and improved readability when dealing with recursive algorithms.

    What data structure is commonly used for implementing memoization?

    A dictionary or an array (list) is commonly utilized as data structures for implementing memoization based on specific problem requirements and constraints.

    Can we apply bottom-up DP instead of top-down DP with recursion?

    Yes, bottom-up dynamic programming (tabulation) involves filling up tables iteratively from base cases rather than breaking down into subproblems like top-down DP (recursion).

    Does every recursive algorithm benefit from using memoization?

    Not all recursive algorithms benefit from using memoization; it is most beneficial for algorithms exhibiting overlapping subproblems to avoid recomputation of already solved subproblems.

    How do I identify opportunities for applying memorisation in problems?

    Look out for repeated computations during recursive calls or multiple paths leading to the same subproblem calculation as indicators that memorisation could be advantageous.


    Mastering LeetCode’s Minimum Falling Path Sum problem with memoziation involves grasping how dynamic programming techniques such as memorisation can optimize computations effectively while reducing time complexity.

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