Unique Paths

Medium Subject: 2-D Dynamic Programming

Time Complexity

O(M * N)

Space Complexity

O(M * N)

Problem Description

Problem Statement

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.

Example 1:

Input: m = 3, n = 7
Output: 28

Optimal Solution

Python

Approach: Dynamic Programming

We can use a 2D array dp where dp[i][j] represents the number of unique paths to reach cell (i, j). The first row and first column will have only 1 unique path (all rights or all downs). For any other cell, dp[i][j] = dp[i-1][j] + dp[i][j-1].

class Solution:
    def uniquePaths(self, m: int, n: int) -> int:
        dp = [[1] * n for _ in range(m)]

        for i in range(1, m):
            for j in range(1, n):
                dp[i][j] = dp[i-1][j] + dp[i][j-1]

        return dp[m-1][n-1]