Version I - n^2 extra space

public class Solution {
    public int minimumTotal(List<List<Integer>> triangle) {
        int n = triangle.size();

        int[][] dp = new int[n][n];

        for (int i = 0; i < triangle.get(n - 1).size(); ++i) {
            dp[n - 1][i] = triangle.get(n - 1).get(i);
        }

        for (int i = n - 2; i >= 0; --i) {
            for (int j = 0; j < triangle.get(i).size(); ++j) {
                dp[i][j] = Math.min(dp[i + 1][j], dp[i + 1][j + 1]) + triangle.get(i).get(j);
            }
        }

        return dp[0][0];
    }
}

Version II - O(n) extra space

public class Solution {
    public int minimumTotal(List<List<Integer>> triangle) {
        if (triangle.size() == 0) return 0;

        int n = triangle.size();

        int[] dp = new int[n + 1];

        for (int i = n - 1; i >= 0; --i) {
            for (int j = 0; j < triangle.get(i).size(); ++j) {
                dp[j] = Math.min(dp[j], dp[j + 1]) + triangle.get(i).get(j);
            }
        }

        return dp[0];
    }
}