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Unique Paths Leetcode Problem 62 [Python Solution]
In this blog post, we’re going to explore the Unique Paths problem, categorized as a 2-D Dynamic Programming challenge on LeetCode.
This problem is considered to be of medium difficulty and is a classic example of dynamic programming.
We’ll provide a Python solution to this problem, optimize it for SEO, and ensure that beginners can follow along with detailed explanations.
Problem Name: Unique Paths
Difficulty: Medium
Category: 2-D Dynamic Programming
Companies: Google
Keyword: Unique Paths LeetCode
Post Title: Unique Paths LeetCode Problem 62 [Python Solution]
Question Link: LeetCode Problem 62
Question: There is a robot on an m x n grid.
The robot is initially located at the top-left corner (i.e., grid[0][0]).
The robot tries to move to the bottom-right corner (i.e., grid[m – 1][n – 1]).
The robot can only move either down or right at any point in time.
Given the two integers m and n, return the number of possible unique paths that the robot can take to reach the bottom-right corner.
The test cases are generated so that the answer will be less than or equal to 2 * 10^9.
Example 1:
Input: m = 3, n = 7
Output: 28
Example 2:
Input: m = 3, n = 2
Output: 3
Explanation: From the top-left corner, there are a total of 3 ways to reach the bottom-right corner:
- Right -> Down -> Down
- Down -> Down -> Right
- Down -> Right -> Down
Constraints:
1 <= m, n <= 100
Problem Overview
The Unique Paths problem presents us with an m x n grid.
The goal is to find the number of unique paths a robot can take to reach the bottom-right corner of the grid, starting from the top-left corner.
The robot can only move either down or right at any given point.
Understanding Constraints
Before we delve into the solution, it’s crucial to understand the constraints of this problem.
The grid’s dimensions are limited to 100×100, and the answer is guaranteed to be less than or equal to 2 * 10^9.
Unique Paths LeetCode Problem Solution
Now, let’s explore the Python solution for the Unique Paths problem.
1. Bruteforce Approach
In the brute-force approach, we can use a recursive algorithm to explore all possible paths from the top-left to the bottom-right corner of the grid.
However, this method can be very inefficient for larger grids, making it impractical for this problem.
2. An Efficient Approach with Dynamic Programming
To efficiently solve this problem, we’ll use dynamic programming.
We can create a 2D array to store the number of unique paths for each cell in the grid.
Here’s the Python code solution:
def uniquePaths(self, m: int, n: int) -> int:
row = [1] * n
for i in range(m - 1):
newRow = [1] * n
for j in range(n - 2, -1, -1):
newRow[j] = newRow[j + 1] + row[j]
row = newRow
return row[0]
This code efficiently computes the number of unique paths using dynamic programming.
Time and Space Complexity
The time complexity of this solution is O(m * n), and the space complexity is O(n).
Reasoning Behind Our Approach
To understand the reasoning behind our efficient approach, consider that we have two choices at any given position in the grid: move down or move right.
By storing and reusing intermediate results, we avoid redundant calculations, making the solution more efficient.
In the code, we create a row and update it iteratively, computing the unique paths row by row until we reach the top-left corner, which contains the final answer.
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Conclusion
In this blog post, we explored the Unique Paths problem, provided an efficient Python solution using dynamic programming, and explained the reasoning behind our approach.
This solution can handle large grid sizes and is an excellent example of using dynamic programming to optimize solutions.
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Happy coding!
