Triangle

26% Accepted
Given a triangle, find the minimum path sum from top to bottom.
Each step you may move to adjacent numbers on the row below.

Have you met this question in a real interview? Yes
Example
For example, given the following triangle

[
     [2],
    [3,4],
   [6,5,7],
  [4,1,8,3]
]
The minimum path sum from top to bottom is 11 (i.e., 2 + 3 + 5 + 1 = 11).

Note

Bonus point if you are able to do this using only O(n) extra space, where n is the total number of rows in the triangle.

Tags Expand

• Dynamic Programming
• 这个问题从bottom往top思考很好想，一直在找最小值(底层只有来自上层的两个选择)
• bottom -> top ， 就能使用分治法递归啦，递归到最底层，也就是从底层开始算，不断返回到上层这一层的值
• 使用动规，因为相当于二维数组，使用sum[][]作为辅助
• state: sum[i][j]表示 i行j列时，从小往上的最小和
• function: sum[i][j] = triangle.get(i).get(j).intValue() + Math.min(sum[i+1][j], sum[i+1][j+1]);
• initialize： 把底层的数先赋初值，sum[n-1][i] = triangle.get(n-1).get(i).intValue();
• answer: sum[0][0]

bottom-up动态规划

public class Solution {
    /**
     * @param triangle: a list of lists of integers.
     * @return: An integer, minimum path sum.
     */
    public int minimumTotal(ArrayList<ArrayList<Integer>> triangle) {
        // write your code here
        int n = triangle.size();
        int[][] sum = new int[n][n];

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

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

        return sum[0][0];
    }


}

分治法

public class Solution {
    /**
     * @param triangle: a list of lists of integers.
     * @return: An integer, minimum path sum.
     */
    public int minimumTotal(ArrayList<ArrayList<Integer>> triangle) {
        // write your code here
        return divideconquer(0, 0, triangle);
    }

    public int divideconquer(int x, int y, ArrayList<ArrayList<Integer>> triangle){
        if (y == triangle.size() || x == triangle.size()) {
            return 0;
        }
        int min = Math.min(divideconquer(x,y+1,triangle),divideconquer(x+1,y+1,triangle));
        return min + triangle.get(y).get(x).intValue();
    }


}