Problem description

Given N, consider a convex N-sided polygon with vertices labelled A[0], A[i], …, A[N-1] in clockwise order.

Suppose you triangulate the polygon into N-2 triangles. For each triangle, the value of that triangle is the product of the labels of the vertices, and the total score of the triangulation is the sum of these values over all N-2 triangles in the triangulation.

Return the smallest possible total score that you can achieve with some triangulation of the polygon.

Example 1:

Input: [1,2,3] Output: 6 Explanation: The polygon is already triangulated, and the score of the only triangle is 6. Example 2:

Input: [3,7,4,5] Output: 144 Explanation: There are two triangulations, with possible scores: 3*7*5 + 4*5*7 = 245, or 3*4*5 + 3*4*7 = 144. The minimum score is 144. Example 3:

Input: [1,3,1,4,1,5] Output: 13 Explanation: The minimum score triangulation has score 1*1*3 + 1*1*4 + 1*1*5 + 1*1*1 = 13.

Solution

1. Notice that in the final triagulation edge (0, N - 1) must belong to a triangle. There are N - 2 options (from 1~N-2) to choose the third point of the triangle.
2. Let DP[i, j] be the minimum cost to triangularization ploygon with edge (i, j and points from i~j). Then \(DP(i, j)=min_{i<k<j} (DP(i, k) + DP(k, j) + A[i]A[k]A[j])\)
3. Base case is when i~j contains only two points. DP(i, j) = 0
4. We can solve above equation using dynamic programming

Below is python implementation

class Solution:
    def minScoreTriangulation(self, A: List[int]) -> int:
        n = len(A)
        DP = [[float('inf') for j in range(n)] for i in range(n)]
        for i in range(0, n - 1):
            DP[i][i + 1] = 0
        
        for l in range(3, n + 1):
            for i in range(0, n + 1 - l):
                j = l + i - 1
                for k in range(i, j):
                    DP[i][j] = min(DP[i][j], DP[i][k] + DP[k][j] + A[i] * A[j] * A[k])
        return DP[0][n - 1]