The problem statement is deceptive by mentioning path from “top” to “bottom”. I was fooled at first and tried to compare all the ways from the top to the bottoms. However, it is impractical since there are O(2^(n-1)) such paths.
I think this is a typical demonstration of the difference between bottom-up and top-down dynamic programming. It would be remarkably easier if we start from bottom and do upward. The solution is very short when the idea is set.

class Solution{
    int minimumTotal(vector<vector<int>>& triangle ){		
		int N = triangle.size();
			return 0;
		vector<int> r = triangle[N-1];
		for(int i= N-2; i>=0; i--){
			for(int j=0; j<=i; j++){
				r[j] = triangle[i][j]+min(r[j],r[j+1]);
		return r[0];
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