# 62. Unique Paths

## Description

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 * 109.

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

## Solutions

Solution 1: Dynamic Programming

We define $f[i][j]$ to represent the number of paths from the top left corner to $(i, j)$, initially $f[0][0] = 1$, and the answer is $f[m - 1][n - 1]$.

Consider $f[i][j]$:

• If $i > 0$, then $f[i][j]$ can be reached by taking one step from $f[i - 1][j]$, so $f[i][j] = f[i][j] + f[i - 1][j]$;
• If $j > 0$, then $f[i][j]$ can be reached by taking one step from $f[i][j - 1]$, so $f[i][j] = f[i][j] + f[i][j - 1]$.

Therefore, we have the following state transition equation:

$f[i][j] = \begin{cases} 1 & i = 0, j = 0 \\ f[i - 1][j] + f[i][j - 1] & \text{otherwise} \end{cases}$

The final answer is $f[m - 1][n - 1]$.

The time complexity is $O(m \times n)$, and the space complexity is $O(m \times n)$. Here, $m$ and $n$ are the number of rows and columns of the grid, respectively.

We notice that $f[i][j]$ is only related to $f[i - 1][j]$ and $f[i][j - 1]$, so we can optimize the first dimension space and only keep the second dimension space, resulting in a time complexity of $O(m \times n)$ and a space complexity of $O(n)$.

• class Solution {
public int uniquePaths(int m, int n) {
int[] f = new int[n];
Arrays.fill(f, 1);
for (int i = 1; i < m; ++i) {
for (int j = 1; j < n; ++j) {
f[j] += f[j - 1];
}
}
return f[n - 1];
}
}

• class Solution {
public:
int uniquePaths(int m, int n) {
vector<int> f(n, 1);
for (int i = 1; i < m; ++i) {
for (int j = 1; j < n; ++j) {
f[j] += f[j - 1];
}
}
return f[n - 1];
}
};

• class Solution:
def uniquePaths(self, m: int, n: int) -> int:
# avoid setting dp[][] to 1 for i==0 or j==0 as initialization
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[-1][-1]

• func uniquePaths(m int, n int) int {
f := make([]int, n+1)
for i := range f {
f[i] = 1
}
for i := 1; i < m; i++ {
for j := 1; j < n; j++ {
f[j] += f[j-1]
}
}
return f[n-1]
}

• function uniquePaths(m: number, n: number): number {
const f: number[] = Array(n).fill(1);
for (let i = 1; i < m; ++i) {
for (let j = 1; j < n; ++j) {
f[j] += f[j - 1];
}
}
return f[n - 1];
}


• /**
* @param {number} m
* @param {number} n
* @return {number}
*/
var uniquePaths = function (m, n) {
const f = Array(n).fill(1);
for (let i = 1; i < m; ++i) {
for (let j = 1; j < n; ++j) {
f[j] += f[j - 1];
}
}
return f[n - 1];
};


• impl Solution {
pub fn unique_paths(m: i32, n: i32) -> i32 {
let (m, n) = (m as usize, n as usize);
let mut f = vec![1; n];
for i in 1..m {
for j in 1..n {
f[j] += f[j - 1];
}
}
f[n - 1]
}
}