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In this approach, we'll use dynamic programming (DP) to achieve the solution. We'll start from the bottom of the triangle and minimize the sum path to the top. We'll keep updating the path sum in a DP array as we move to the top row.
Time Complexity: O(n^2), where n is the number of rows in the triangle.
Space Complexity: O(n), due to the use of the array dp.
1def minimumTotal(triangle):
2 dp = triangle[-1]
3 for row in range(len(triangle) - 2, -1, -1):
4 for col in range(len(triangle[row])):
5 dp[col] = triangle[row][col] + min(dp[col], dp[col + 1])
6 return dp[0]
7
8triangle = [[2], [3, 4], [6, 5, 7], [4, 1, 8, 3]]
9print(minimumTotal(triangle))The Python function minimumTotal utilizes dynamic programming to track the minimum path sum from bottom to top. The input triangle gets updated iteratively via the variable dp which represents the current base row in the triangle.
This approach uses a top-down strategy with memoization to store already computed results, avoiding redundant calculations. Each recursive call stores the minimum path sum starting from the current index down to the base of the triangle.
Time Complexity: O(n^2) (due to memoization)
Space Complexity: O(n^2) for memoization storage.
1#include <vector>
#include <algorithm>
#include <cstring>
#include <iostream>
using namespace std;
int memo[201][201];
class Solution {
public:
int dfs(int row, int col, vector<vector<int>>& triangle) {
if (row == triangle.size()) return 0;
if (memo[row][col] != -1) return memo[row][col];
int left = dfs(row + 1, col, triangle);
int right = dfs(row + 1, col + 1, triangle);
memo[row][col] = triangle[row][col] + min(left, right);
return memo[row][col];
}
int minimumTotal(vector<vector<int>>& triangle) {
memset(memo, -1, sizeof(memo));
return dfs(0, 0, triangle);
}
};
int main() {
Solution sol;
vector<vector<int>> triangle = {{2}, {3, 4}, {6, 5, 7}, {4, 1, 8, 3}};
cout << sol.minimumTotal(triangle) << endl;
return 0;
}C++ implementation uses recursion with memoization to compute minimum path sums from top to leaf nodes. dfs efficiently computes in bottom paths using the array memo to eliminate redundant calculations.