Unique Paths

Problem

https://leetcode.com/problems/unique-paths/

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:sup:`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:
1. Right -> Down -> Down
2. Down -> Down -> Right
3. Down -> Right -> Down

Constraints:

1 <= m, n <= 100

Pattern

Math, Dynamic Programming, Combinatorics

Approaches

Dynamic Programming

Code

def uniquePaths(m: int, n: int) -> int:
    paths = [[0] * n for _ in range(m)]

    for i in range(m):
        for j in range(n):
            if i == 0 and j == 0:
                paths[0][0] = 1
            elif i == 0:
                paths[i][j] = paths[i][j - 1]
            elif j == 0:
                paths[i][j] = paths[i - 1][j]
            else:
                paths[i][j] = paths[i - 1][j] + paths[i][j - 1]

    return paths[-1][-1]

Test

>>> from unique_paths__dynamic_programming import uniquePaths
>>> uniquePaths(3, 7)
28
>>> uniquePaths(3, 2)
3

unique_paths__dynamic_programming.uniquePaths(m: int, n: int) → int