# How to solve Leetcode 63. Unique Paths II

## Dynamic programming in place

### Problem statement

You are given an `m x n`

integer array `grid`

. There is a robot 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.

An obstacle and space are marked as `1`

or `0`

respectively in `grid`

. A path that the robot takes cannot include any square that is an obstacle.

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: obstacleGrid = [[0,0,0],[0,1,0],[0,0,0]]
Output: 2
Explanation: There is one obstacle in the middle of the 3x3 grid above.
There are two ways to reach the bottom-right corner:
1. Right -> Right -> Down -> Down
2. Down -> Down -> Right -> Right
```

#### Example 2

```
Input: obstacleGrid = [[0,1],[0,0]]
Output: 1
```

#### Constraints

`m == obstacleGrid.length`

.`n == obstacleGrid[i].length`

.`1 <= m, n <= 100`

.`obstacleGrid[i][j]`

is`0`

or`1`

.

### Solution: Dynamic programming in place

Let us find the relationship between the positions.

If there is no obstacle at position `(row = i, col = j)`

, the number of paths `np[i][j]`

that the robot can take to reach this position is:

```
np[i][j] = np[i - 1][j] + np[i][j - 1]
```

As long as there is no obstacle in the first row,

`np[0][j] = 1`

. Otherwise,`np[0][k] = 0`

for all`k >= j0`

, where`(0, j0)`

is the position of the first obstacle in the first row.Similarly, as long as there is no obstacle in the first column,

`np[i][0] = 1`

. Otherwise,`np[k][0] = 0`

for all`k >= i0`

, where`(i0, 0)`

is the position of the first obstacle in the first column.

#### Code

```
#include <vector>
#include <iostream>
using namespace std;
int uniquePathsWithObstacles(vector<vector<int>>& obstacleGrid) {
const int row = obstacleGrid.size();
const int col = obstacleGrid[0].size();
vector<vector<int>> np(row, vector<int>(col, 0));
for (int i = 0; i < row && obstacleGrid[i][0] == 0; i++) {
np[i][0] = 1;
}
for (int j = 0; j < col && obstacleGrid[0][j] == 0; j++) {
np[0][j] = 1;
}
for (int i = 1; i < row; i++) {
for (int j = 1; j < col; j++) {
if (obstacleGrid[i][j] == 0) {
np[i][j] = np[i - 1][j] + np[i][j - 1];
}
}
}
return np[row - 1][col - 1];
}
int main() {
vector<vector<int>> obstacleGrid = {{0,0,0},{0,1,0},{0,0,0}};
cout << uniquePathsWithObstacles(obstacleGrid) << endl;
obstacleGrid = {{0,1},{0,0}};
cout << uniquePathsWithObstacles(obstacleGrid) << endl;
}
```

```
Output:
2
1
```

#### Complexity

Runtime:

`O(m*n)`

, where`m x n`

is the size of the`grid`

.Extra space:

`O(m*n)`

.

### References

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