The Problem:

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

For example, given the following triangle


The minimum path sum from top to bottom is 11 (i.e., 2 + 3 + 5 + 1 = 11).

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.


  1. The main idea is traversing the list backwards, which means start from bottom of the triangle. For each level up, store the result of min path to reach this node from lower level.
  2. It is a DP problem, but the previous results for one level can be updated in place.

Java Solution

public class Solution {
    public int minimumTotal(List<List<Integer>> triangle) {
            return triangle.get(0).get(0);
        int rows = triangle.size();
        int sum[] = new int[triangle.get(rows-1).size()];
        for(int i = 0; i < triangle.get(rows-1).size(); i++ ){
            sum[i] = triangle.get(rows-1).get(i);
        for(int i = rows-2; i>= 0; i--){
            for(int j = 0; j <  triangle.get(i).size(); j++ ){
                sum[j] = triangle.get(i).get(j) + Math.min(sum[j], sum[j +1]);
        return sum[0];

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