What is the time complexity of Out of Boundary Paths?

The optimal solution has a time complexity of O(maxMove * m * n).

What is the space complexity of Out of Boundary Paths?

The space complexity is O(m * n).

What data structures are used to solve Out of Boundary Paths?

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

What are the key insights for solving Out of Boundary Paths?

Insight 1: Dynamic programming is suitable because the problem involves finding the number of paths, and the result for a given cell and move count depends on the results of its neighboring cells with one less move. Insight 2: We can use a 2D array to store the number of ways to reach each cell in the grid after a certain number of moves.

Which companies ask LeetCode 576. Out of Boundary Paths in interviews?

This problem is frequently asked at Baidu.

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Out of Boundary Paths - LeetCode 576 Solution

Out of Boundary Paths - Complete Solution Guide

Out of Boundary Paths is LeetCode problem 576, 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

There is an m x n grid with a ball. The ball is initially at the position [startRow, startColumn] . You are allowed to move the ball to one of the four adjacent cells in the grid (possibly out of the grid crossing the grid boundary). You can apply at most maxMove moves to the ball. Given the five integers m , n , maxMove , startRow , startColumn , return the number of paths to move the ball out of the grid boundary. Since the answer can be very large, return it modulo 10 9 + 7 . Example 1: Input

Detailed Explanation

The problem asks us to find the number of paths a ball can take to go out of bounds of an m x n grid, given a starting position [startRow, startColumn] and a maximum number of moves, maxMove. The ball can move to any of its four adjacent cells (up, down, left, right) in each move. We need to return the count of paths that lead the ball out of the grid's boundaries, modulo 10^9 + 7.

Solution Approach

The solution uses dynamic programming to solve this problem. We maintain a 2D array `dp` where `dp[r][c]` represents the number of ways to reach cell (r, c) from the starting position after a certain number of moves. We iteratively update this array for each move, considering the contributions from its neighbors. If a move takes the ball out of bounds, we increment a counter. The final count is the number of paths that lead the ball out of bounds after `maxMove` moves.

Step-by-Step Algorithm

Step 1: Initialize a 2D array `dp` of size m x n with all values set to 0. Set `dp[startRow][startColumn] = 1`, indicating that there is one way to reach the starting cell with zero moves.
Step 2: Initialize a counter `count` to 0. This counter will store the number of paths that lead the ball out of bounds.
Step 3: Iterate `maxMove` times. In each iteration, create a temporary 2D array `temp` of size m x n, initialized with all values set to 0.
Step 4: Iterate through each cell (r, c) in the `dp` array.
Step 5: If `dp[r][c] > 0`, it means there is at least one way to reach this cell. Consider all four possible moves (up, down, left, right) from this cell.
Step 6: For each move, calculate the new row `nr` and new column `nc`.
Step 7: If `(nr, nc)` is within the grid boundaries (0 <= nr < m and 0 <= nc < n), update `temp[nr][nc]` by adding `dp[r][c]` to it, modulo (10^9 + 7).
Step 8: If `(nr, nc)` is out of bounds, increment `count` by `dp[r][c]`, modulo (10^9 + 7).
Step 9: After iterating through all cells in the `dp` array, update `dp` with the values from the `temp` array. This represents the state of the grid after one more move.
Step 10: After completing all `maxMove` iterations, return the final value of `count`.

Key Insights

Insight 1: Dynamic programming is suitable because the problem involves finding the number of paths, and the result for a given cell and move count depends on the results of its neighboring cells with one less move.
Insight 2: We can use a 2D array to store the number of ways to reach each cell in the grid after a certain number of moves.
Insight 3: The core idea is to iteratively update the DP array, where each iteration represents a move. We determine the number of paths that lead to out-of-bounds in each move and accumulate this value.

Complexity Analysis

Time Complexity: O(maxMove * m * n)

Space Complexity: O(m * n)

Topics

This problem involves: Dynamic Programming.

Companies

Asked at: Baidu.

Out of Boundary Paths | LeetCode 576

Medium

Dynamic Programming

Time: O(maxMove * m * n)Space: O(m * n)

Solution Code

Available in:

class Solution {
    public int findPaths(int m, int n, int maxMove, int startRow, int startColumn) {
        int MOD = (int)1e9 + 7;
        
        int[][] dp = new int[m][n];
        dp[startRow][startColumn] = 1;
        
        int count = 0;
        
        for (int i = 0; i < maxMove; i++) {
            int[][] temp = new int[m][n];
            for (int r = 0; r < m; r++) {
                for (int c = 0; c < n; c++) {
                    if (dp[r][c] > 0) {
                        int[][] directions = {{0, 1}, {0, -1}, {1, 0}, {-1, 0}};
                        for (int[] dir : directions) {
                            int nr = r + dir[0];
                            int nc = c + dir[1];
                            
                            if (0 <= nr && nr < m && 0 <= nc && nc < n) {
                                temp[nr][nc] = (temp[nr][nc] + dp[r][c]) % MOD;
                            } else {
                                count = (count + dp[r][c]) % MOD;
                            }
                        }
                    }
                }
            }
            dp = temp;
            
        }
        
        return count;
    }
}

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Problem Info

Description

Problem 576: Out of Boundary Paths - Detailed explanation and solution approaches.

Topics

Dynamic Programming

Companies

Baidu

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:576

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

10 steps

1
Step 1: Initialize a 2D array `dp` of size m x n with all values set to 0. Set `dp[startRow][startColumn] = 1`, indicating that there is one way to reach the starting cell with zero moves.
2
Step 2: Initialize a counter `count` to 0. This counter will store the number of paths that lead the ball out of bounds.
3
Step 3: Iterate `maxMove` times. In each iteration, create a temporary 2D array `temp` of size m x n, initialized with all values set to 0.
4
Step 4: Iterate through each cell (r, c) in the `dp` array.
5
Step 5: If `dp[r][c] > 0`, it means there is at least one way to reach this cell. Consider all four possible moves (up, down, left, right) from this cell.
6
Step 6: For each move, calculate the new row `nr` and new column `nc`.
7
Step 7: If `(nr, nc)` is within the grid boundaries (0 <= nr < m and 0 <= nc < n), update `temp[nr][nc]` by adding `dp[r][c]` to it, modulo (10^9 + 7).
8
Step 8: If `(nr, nc)` is out of bounds, increment `count` by `dp[r][c]`, modulo (10^9 + 7).
9
Step 9: After iterating through all cells in the `dp` array, update `dp` with the values from the `temp` array. This represents the state of the grid after one more move.
10
Step 10: After completing all `maxMove` iterations, return the final value of `count`.

💡

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(maxMove * m * n)

💾

Space Complexity

O(m * n)

📝

Why?

The time complexity is derived directly from the nested loops. The outer loop runs `maxMove` times, and the two inner loops iterate through the m x n grid. The space complexity results from storing the `dp` and `temp` arrays, both of which require O(m*n) space.

📖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

Recursion with Memoization

A recursive approach with memoization can also solve this problem. The recursive function would take the current row, current column, and remaining moves as input and return the number of paths to go out of bounds. Memoization is used to avoid recomputing the same subproblems.

Time: O(maxMove * m * n)Space: O(maxMove * m * n)

Trade-offs: The recursive approach with memoization has the same time and space complexity as the dynamic programming approach, but it may be less efficient due to the overhead of recursion. However, it can be more intuitive to implement for some programmers.

✨

Key Insights

1Insight 1: Dynamic programming is suitable because the problem involves finding the number of paths, and the result for a given cell and move count depends on the results of its neighboring cells with one less move.
2Insight 2: We can use a 2D array to store the number of ways to reach each cell in the grid after a certain number of moves.
3Insight 3: The core idea is to iteratively update the DP array, where each iteration represents a move. We determine the number of paths that lead to out-of-bounds in each move and accumulate this value.

📚

Key Terms Explained

For Students

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: `maxMove = 0`. In this case, the ball cannot move, so if the ball starts out of bounds, the result would be 1; otherwise, the result is 0. The provided solution correctly handles this because the loop doesn't execute if `maxMove` is zero, and the initial count remains zero.
Edge case 2: `m = 1, n = 1, maxMove > 0`. The ball will always go out of bounds on the first move. The solution correctly identifies this and counts the number of paths from the starting location.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to take the modulo (10^9 + 7) at each step. If the modulo operation is omitted, integer overflow can occur, leading to incorrect results.

✗Common mistake 2: Not creating a separate `temp` array to store the intermediate results for each move. Modifying the `dp` array directly during the iteration can lead to incorrect counts because the contributions from different cells within the same move can overwrite each other.

✗Common mistake 3: Not handling the edge cases where maxMove is 0 appropriately, or when the grid dimensions are small.

✅

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: Pathfinding in a restricted environment. Imagine a robot navigating a warehouse with certain boundaries and a limited battery life (maxMove). This algorithm can help determine the number of ways the robot can 'escape' (go out of bounds) the warehouse given its starting location and battery limitations.

🏦Real-world scenario 2: Modeling particle diffusion. The grid could represent a space, and the ball represents a particle. The algorithm simulates how many ways the particle can escape the space after a certain number of random movements.

💡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

Start with recursion, then memoize. The hardest part is identifying the state. Ask yourself: "What information do I need to make the current decision?"

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Forgetting base cases or initializing the DP array incorrectly. Start by solving the smallest subproblems.

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 7, 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

❓

Frequently Asked Questions

How do I identify if a problem needs dynamic programming?

DP problems typically have: 1) Overlapping subproblems (same calculation repeated), 2) Optimal substructure (optimal solution uses optimal solutions to subproblems). Look for keywords like "minimum/maximum," "count ways," "is it possible," or "longest/shortest."

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Out of Boundary Paths"?

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

🤖 Need help with this problem?