Triangle

Important:

The hardest part is to define the state(here is f[x][y]) and its function(the relationship between them)

public class Solution {
    /**
     * @param triangle: a list of lists of integers.
     * @return: An integer, minimum path sum.
     */
    public int minimumTotal(int[][] triangle) {
        // write your code here
        if (triangle == null || triangle.length == 0) {
            return 0;
        } 
        
        if (triangle[0] == null || triangle[0].length == 0) {
            return 0;
        }
        
        int n = triangle.length;
        
        //initiallize 
        //f[x][y]: minimum path value from (0,0) to (x, y)
        int[][] f = new int[n][n];
        //top value
        f[0][0] = triangle[0][0];  
        //two sides
        for (int i = 1; i < n; i++) {
            f[i][0] = f[i - 1][0] + triangle[i][0];
            f[i][i] = f[i - 1][i - 1] + triangle[i][i];
        }
        
        //Top to bottom
        for(int i = 1; i < n; i++) {
            for(int j = 1; j < i; j++) {
                f[i][j] = Math.min(f[i - 1][j - 1], f[i - 1][j]) + triangle[i][j];
            }
        }
        
        //find the minimum in the bottom
        int minimum = f[n - 1][0];
        for(int i = 1; i < n; i++) {
            if (minimum > f[n - 1][i]) {
                minimum = f[n - 1][i];
            }
        }
        return minimum;
    }
}