LeetCode - Unique Paths

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.

Solution1:

2维DP：f(i, j) = f(i-1, j) + f(i, j-1)。

初始化：f(0, j) = 1, 0<=j<n;  f(i, 0) = 1, 0<=i<m

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

 

Solution2:

1维DP：f(i) = f(i) + f(i-1)。

初始化：f(0) = 1

public int uniquePaths(int m, int n) {
    int[] f = new int[n];
    f[0] = 1;
    for(int i=0; i<m; i++) {
        for(int j=0; j<n; j++) {
            f[j] = f[j] + (j>0?f[j-1]:0);
        }
    }
    return f[n-1];
}