# How To Solve Leetcode 62. Unique Paths

## Problem statement

A robot is located at the top-left corner of a `m x n` grid (marked 'Start' in the diagram below).

The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).

How many possible unique paths are there?

### Example 1

![62_robot_maze.png](https://cdn.hashnode.com/res/hashnode/image/upload/v1655235331734/d1ltc0Fxt.png align="left")

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

### Example 2

```plaintext
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
```

### Example 3

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

### Example 4

```plaintext
Input: m = 3, n = 3
Output: 6
```

### Constraints

* `1 <= m, n <= 100`.
    
* It's guaranteed that the answer will be less than or equal to `2*10^9`.
    

## Solution 1: Recursive

At each point, the robot has two ways of moving: right or down. Let `P(m,n)` is the wanted result. Then you have a recursive relationship:

```plaintext
P(m,n) = P(m-1, n) + P(m, n-1)
```

If the grid has only one row or only one column, then there is only one possible path.

```plaintext
P(1, n) = P(m, 1) = 1.
```

We have a recursive implementation.

### Code

```cpp
#include <iostream>
#include <vector>
using namespace std;
int uniquePaths(int m, int n) {
    if (m == 1 || n == 1) {
        return 1;
    }
    return uniquePaths(m - 1, n) + uniquePaths(m, n - 1);
}
int main() {
    std::cout << uniquePaths(3,7) << std::endl;
    std::cout << uniquePaths(7,3) << std::endl;
    std::cout << uniquePaths(3,2) << std::endl;
    std::cout << uniquePaths(3,3) << std::endl;
}
```

```plaintext
Output:
28
28
3
6
```

### Complexity

* Runtime: `O(2^m + 2^n)`, where `m*n` is the size of the grid.
    
* Extra space: `O(2^m + 2^n)`.
    

## Solution 2: Dynamic programming

The recursive implementation repeats a lot of computations.

For example, `uniquePaths(2,2)` was recomputed in both `uniquePaths(2,3)` and `uniquePaths(3,2)` when you compute `uniquePaths(3,3)`.

One way of storing what has been computed is by using dynamic programming.

### Code

```cpp
#include <iostream>
#include <vector>
using namespace std;
int uniquePaths(int m, int n) {
    vector<vector<int> > dp(m, vector<int>(n,1));
    for (int i = 1; i < m; i++) {
        for (int j = 1; j < n; j++) {
            dp[i][j] = dp[i - 1][j] + dp[i][j - 1];
        }
    }
    return dp[m - 1][n - 1];
}
int main() {
    std::cout << uniquePaths(3,7) << std::endl;
    std::cout << uniquePaths(7,3) << std::endl;
    std::cout << uniquePaths(3,2) << std::endl;
    std::cout << uniquePaths(3,3) << std::endl;
}
```

```plaintext
Output:
28
28
3
6
```

### Complexity

* Runtime: `O(m*n)`, where `m*n` is the size of the grid.
    
* Extra space: `O(m*n)`.
    

## Solution 3: Reduced dynamic programming

You can rephrase the relationship inside the loop like this:

> "new value" = "old value" + "previous value";

Then you do not have to store all values of all rows.

### Code

```cpp
#include <iostream>
#include <vector>
using namespace std;
int uniquePaths(int m, int n) {
    vector<int> dp(n, 1);
    for (int i = 1; i < m; i++) {
        for (int j = 1; j < n; j++) {
            dp[j] += dp[j - 1];
        }
    }
    return dp[n - 1];
}
int main() {
    std::cout << uniquePaths(3,7) << std::endl;
    std::cout << uniquePaths(7,3) << std::endl;
    std::cout << uniquePaths(3,2) << std::endl;
    std::cout << uniquePaths(3,3) << std::endl;
}
```

```plaintext
Output:
28
28
3
6
```

### Complexity

* Runtime `O(m*n)`.
    
* Memory `O(n)`.
    

## Final thought

I am wondering if there is some mathematics behind this problem. Please share your finding if you find a formula for the solution to this problem.

## References

* [https://leetcode.com/problems/unique-paths/](https://leetcode.com/problems/unique-paths/)
    
* [https://www.leetsolve.com/62-unique-paths](https://www.leetsolve.com/62-unique-paths)
    

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