What is the time complexity of Minimum Falling Path Sum?

The optimal solution has a time complexity of O(n^2).

What data structures are used to solve Minimum Falling Path Sum?

This problem uses Array, Dynamic Programming, Matrix. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Minimum Falling Path Sum?

Insight 1: Dynamic programming can be used to efficiently solve this problem by building up the minimum falling path sum from the top row to the bottom row. Insight 2: The minimum falling path sum at a cell (row, col) is the cell's value plus the minimum falling path sum from any of its possible parent cells in the previous row (row - 1, col - 1), (row - 1, col), or (row - 1, col + 1).

Which companies ask LeetCode 931. Minimum Falling Path Sum in interviews?

This problem is frequently asked at Dream11, Intuit.

JavaScript is required for the full interactive experience. However, all content is accessible without JavaScript.

Minimum Falling Path Sum - LeetCode 931 Solution

Minimum Falling Path Sum - Complete Solution Guide

Minimum Falling Path Sum is LeetCode problem 931, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given an n x n array of integers matrix , return the minimum sum of any falling path through matrix . A falling path starts at any element in the first row and chooses the element in the next row that is either directly below or diagonally left/right. Specifically, the next element from position (row, col) will be (row + 1, col - 1) , (row + 1, col) , or (row + 1, col + 1) . Example 1: Input: matrix = [[2,1,3],[6,5,4],[7,8,9]] Output: 13 Explanation: There are two falling paths with a minimum su

Detailed Explanation

The problem asks us to find the minimum sum of a 'falling path' through a given square matrix. A falling path starts from any cell in the first row and moves to the next row, choosing a cell directly below or diagonally left or right. We need to find the path that results in the smallest possible sum when all the cell values along the path are added together. The input is an n x n matrix of integers, and the output is the minimum falling path sum.

Solution Approach

The solution uses dynamic programming. We iterate through the matrix from the second row to the last row. For each cell, we calculate the minimum falling path sum by adding the current cell's value to the minimum of its possible 'parent' cells in the previous row. The possible parent cells are the cell directly above, the cell diagonally left, and the cell diagonally right. We handle edge cases where the cell is on the left or right boundary of the matrix. After processing all the cells, the last row of the matrix will contain the minimum falling path sums for paths ending at each cell in that row. Finally, we find the minimum value in the last row, which is the overall minimum falling path sum.

Step-by-Step Algorithm

Step 1: Initialize the dynamic programming process by starting from the second row (index 1) of the matrix.
Step 2: Iterate through each cell in the current row.
Step 3: For each cell, determine its possible 'parent' cells in the previous row: directly above, diagonally left, and diagonally right. Handle boundary conditions (leftmost and rightmost columns) appropriately.
Step 4: Calculate the minimum falling path sum to the current cell by adding the current cell's value to the minimum falling path sum of its valid parent cells.
Step 5: Store the calculated minimum falling path sum back into the current cell in the matrix. This overwrites the original value with the minimum path sum up to that point.
Step 6: After processing all rows, the last row of the matrix will contain the minimum falling path sums for all possible falling paths ending in that row.
Step 7: Find the minimum value in the last row. This value represents the overall minimum falling path sum through the matrix.
Step 8: Return the overall minimum falling path sum.

Key Insights

Insight 1: Dynamic programming can be used to efficiently solve this problem by building up the minimum falling path sum from the top row to the bottom row.
Insight 2: The minimum falling path sum at a cell (row, col) is the cell's value plus the minimum falling path sum from any of its possible parent cells in the previous row (row - 1, col - 1), (row - 1, col), or (row - 1, col + 1).
Insight 3: We can optimize space by modifying the input matrix in-place to store the intermediate minimum falling path sums.

Complexity Analysis

Time Complexity: O(n^2)

Space Complexity: O(1)

Topics

This problem involves: Array, Dynamic Programming, Matrix.

Companies

Asked at: Dream11, Intuit.

Minimum Falling Path Sum | LeetCode 931

Medium

Array Dynamic Programming Matrix

Time: O(n^2)Space: O(1)

Solution Code

Available in:

class Solution {
    public int minFallingPathSum(int[][] matrix) {
        int n = matrix.length;

        for (int i = 1; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (j == 0) {
                    matrix[i][j] += Math.min(matrix[i - 1][j], matrix[i - 1][j + 1]);
                } else if (j == n - 1) {
                    matrix[i][j] += Math.min(matrix[i - 1][j], matrix[i - 1][j - 1]);
                } else {
                    matrix[i][j] += Math.min(matrix[i - 1][j - 1], Math.min(matrix[i - 1][j], matrix[i - 1][j + 1]));
                }
            }
        }
        
        int min = matrix[n - 1][0];
        for (int i = 1; i < n; i++) {
            min = Math.min(min, matrix[n - 1][i]);
        }
        return min;
    }
}

View Full Solution on GitHubGitHub Solution View on LeetCodeLeetCode

Problem Info

Description

Problem 931: Minimum Falling Path Sum - Detailed explanation and solution approaches.

Topics

Array Dynamic Programming Matrix

Companies

Dream11Intuit

Stats

Difficulty:Medium

Languages:4

Companies:2

Problem #:931

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

8 steps

1
Step 1: Initialize the dynamic programming process by starting from the second row (index 1) of the matrix.
2
Step 2: Iterate through each cell in the current row.
3
Step 3: For each cell, determine its possible 'parent' cells in the previous row: directly above, diagonally left, and diagonally right. Handle boundary conditions (leftmost and rightmost columns) appropriately.
4
Step 4: Calculate the minimum falling path sum to the current cell by adding the current cell's value to the minimum falling path sum of its valid parent cells.
5
Step 5: Store the calculated minimum falling path sum back into the current cell in the matrix. This overwrites the original value with the minimum path sum up to that point.
6
Step 6: After processing all rows, the last row of the matrix will contain the minimum falling path sums for all possible falling paths ending in that row.
7
Step 7: Find the minimum value in the last row. This value represents the overall minimum falling path sum through the matrix.
8
Step 8: Return the overall minimum falling path sum.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n^2)

🐢 Quadratic - Gets slow with large inputs. Consider optimization!

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The nested loops iterate through each cell of the n x n matrix, resulting in O(n^2) operations. Since the algorithm modifies the input matrix directly and uses only a constant amount of extra memory for variables, the space complexity is O(1).

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Dynamic Programming with Separate DP Table

Instead of modifying the input matrix, we can create a separate DP table of the same dimensions to store the minimum falling path sums. This approach doesn't modify the input.

Time: O(n^2)Space: O(n^2)

Trade-offs: This approach uses O(n^2) extra space but preserves the original matrix. It might be preferred if the original matrix needs to be retained for later use.

Recursion with Memoization

We can solve this using recursion with memoization. Define a recursive function that finds the minimum path sum starting from a given cell. Memoize the results to avoid redundant calculations.

Time: O(n^2)Space: O(n^2) due to the memoization table and recursion stack.

Trade-offs: Recursion with memoization can be more intuitive for some, but might be less efficient than iterative DP in some cases due to function call overhead. The space complexity is O(n^2) due to the memoization table.

✨

Key Insights

1Insight 1: Dynamic programming can be used to efficiently solve this problem by building up the minimum falling path sum from the top row to the bottom row.
2Insight 2: The minimum falling path sum at a cell (row, col) is the cell's value plus the minimum falling path sum from any of its possible parent cells in the previous row (row - 1, col - 1), (row - 1, col), or (row - 1, col + 1).
3Insight 3: We can optimize space by modifying the input matrix in-place to store the intermediate minimum falling path sums.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: Single-element matrix. The algorithm will still work correctly since the loop won't execute, and the first (and only) element is returned.
Edge case 2: Matrix with negative numbers. The algorithm correctly handles negative numbers, as it aims to minimize the sum regardless of the signs of the elements.
Edge case 3: The leftmost and rightmost columns need special handling because they don't have both left and right diagonal neighbors in the row above. The solution addresses this using conditional statements.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to handle the edge cases where the cell is on the left or right boundary of the matrix. This can lead to index out-of-bounds errors or incorrect results.

✗Common mistake 2: Not correctly finding the minimum value in the last row after the DP process. Make sure to iterate through all the elements in the last row to find the smallest value.

✗Common mistake 3: Incorrectly implementing the dynamic programming update rule, leading to incorrect path sums. Ensure to correctly consider all valid parent cells.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Path planning in robotics. Finding the most energy-efficient path for a robot to traverse a grid-like environment where each cell represents energy consumption.

🏦Real-world scenario 2: Image processing. Finding the optimal 'seam' in an image for content-aware image resizing (seam carving). Each pixel's 'energy' can be represented by its gradient, and the minimum energy path is found using a similar dynamic programming approach.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: July 2, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 13

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 6

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJun 29

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

Should I use top-down (memoization) or bottom-up (tabulation)?

Start with top-down (recursion + memoization) as it directly maps to the recursive thinking and handles only needed subproblems. Convert to bottom-up for better space efficiency and to avoid stack overflow. Bottom-up is often faster due to no recursion overhead.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Minimum Falling Path Sum"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether dynamic programming techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?