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?

Above is a 3 x 7 grid. How many possible unique paths are there?

**Note:**

*m*and

*n*will be at most 100.

**Understand the problem:**

The problem asks for how many unique paths start from top-left of the matrix to the bottom-right.

**Recursive Solution:**

The most straight-forward solution is to use recursion. For each node, note that we can only go down or go right.

**Code (Java):**

public class Solution { public int uniquePaths(int m, int n) { if (m == 0 || n == 0) { return 0; } return uniquePathsHelper(m, n); } private int uniquePathsHelper(int m, int n) { if (m == 1 || n == 1) { return 1; } return uniquePathsHelper(m - 1, n) + uniquePathsHelper(m, n - 1); } }

**Discussion:**

We can draw an recursion tree to deeply understand the problem. Note that why m == 1 or n == 1, we return 1. That is because when we reach the bottom or right-most ends, there will be only one solution available. Note that we can only go down or right. For example, there is an 3 x 3 matrix, the recursion tree would look like:

(3, 3)

/ \

(2, 3) (3, 2)

/ \ / \

(1,3) (2,2) (2, 2) (2, 1)

/\ /\

(1,2) (2,1) (1,2) (2,1)

So we can see that the total number of unique paths is 6. We can also clearly see that calculating (2, 2) is redundant. So it is naturally to think about a DP solution.

**DP Solution:**

The crux of the Java solution is to define a transit equation. We define dp[i][j] means the total number of unique paths from dp[0][0] to dp[i][j]. So dp[i][j] = dp[i - 1][j] + dp[i][j - 1]. So we can only need to check about the last element, which is dp[m-1][n-1].

**Code (Java):**

public class Solution { public int uniquePaths(int m, int n) { if (m == 0 || n == 0) { return 0; } int[][] dp = new int[m][n]; for (int i = 0; i < n; i++) { dp[0][i] = 1; } for (int i = 0; i < m; i++) { dp[i][0] = 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]; } }

**Discussion:**

The time complexity is O(m * n), linear to the size of the matrix. The space complexity is O(m * n) since we allocate a dp matrix.

**A space-optimal solution:**

There is another O(n) space solution. The trick is when we do dp[i][j] = dp[i - 1][j] + dp[i][j - 1], we only need to maintain the size of columns array, where as dp[j] means the unique path from 0 to column j, in the current row. For the dp[j] = dp[j] + dp[j - 1], where as dp[j] in the right of the equation is the dp[i - 1][j] in the previous solution. In this way, we don't need to maintain the rows.

**Code (Java):**

public class Solution { public int uniquePaths(int m, int n) { if (m == 0 || n == 0) { return 0; } int[] dp = new int[n]; dp[0] = 1; for (int i = 0; i < m; i++) { for (int j = 1; j < n; j++) { dp[j] = dp[j] + dp[j - 1]; } } return dp[n - 1]; } }

**Another DP Solution:**

Actually a more straight-forward DP solution is directly changed from the recursive solution. In the recursive solution, we just need to cache the intermediate results. It is very similar to the Fibonacci Series problems.

**Code (Java):**

public class Solution { public int uniquePaths(int m, int n) { if (m == 0 || n == 0) { return 0; } int[][] dp = new int[m + 1][n + 1]; return uniquePathsHelper(m, n, dp); } private int uniquePathsHelper(int m, int n, int[][] dp) { if (m == 1 || n == 1) { return 1; } if (dp[m][n] != 0) { return dp[m][n]; } dp[m][n] = uniquePathsHelper(m - 1, n, dp) + uniquePathsHelper(m, n - 1, dp); return dp[m][n]; } }

**Summary:**

This is a very classic recursion and DP problem. Try to understand the details of the recursion solution first. Understand why and how there are duplicated computations. Then go to the DP solution. At least understand the first DP solution which is classical and most commonly used in many other DP problems.

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