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?

Notice

m and n will be at most 100

Solution: Search -> Memory Search -> DP

Search:

public int uniquePaths(int m, int n) {
    if(m <= 0 || n <= 0) {
        return 0;
    }
    int[] res = new int[1];
    res[0] = search(1, 1, res, m, n);
    return res[0];
}
public int search(int m, int n, int[] res, int row, int col) {
    if(m > row || n > col) {
        return 0;
    }
    if(m == row && n == col) {
        return 1;
    }
    return search(m + 1, n, res, row, col) + search(m, n + 1, res, row, col);
}

Memory Search:

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 < m; i++) {
        for(int j = 0; j < n; j++) {
            dp[i][j] = Integer.MAX_VALUE;
        }
    }

    return search(1, 1, m, n, dp);
}
public int search(int m, int n, int row, int col, int[][] dp) {
    if(m > row || n > col) {
        return 0;
    }
    if(m == row && n == col) {
        return 1;
    }
    if(dp[m - 1][n - 1] != Integer.MAX_VALUE) {
        return dp[m - 1][n - 1];
    }
    dp[m - 1][n - 1] = search(m + 1, n, row, col, dp) + search(m, n + 1, row, col, dp);
    return dp[m - 1][n - 1];
}

DP:

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 < m; i++) {
        dp[i][0] = 1;
    }
    for(int i = 0; i < n; i++) {
        dp[0][i] = 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];
}