Optimal Matrix Multiplication

Given two matrices with dimensions $n_1 \times n_2$ and $n_2 \times n_3$ , the cost of the usual multiplication algorithm is $\Theta(n_1 n_2 n_3)$ . For simplicity, let us consider that the cost is exactly $n_1 n_2 n_3$ .

Suppose that we must compute $M_1 \times \dots \times M_m$ , where every $M_i$ is an $n_i \times n_{i+1}$ matrix. Since the product of matrices is associative, we can choose the multiplication order. For example, to compute $M_1 \times M_2 \times M_3 \times M_4$ , we could either choose $(M_1 \times M_2) \times (M_3 \times M_4)$ , with cost $n_1 n_2 n_3 + n_3 n_4 n_5 + n_1 n_3 n_5$ , or either choose $M_1 \times ((M_2 \times M_3) \times M_4)$ , with cost $n_2 n_3 n_4 + n_2 n_4 n_5 + n_1 n_2 n_5$ , or three other possible orders.

Write a program to find the minimum cost of computing $M_1 \times \dots \times M_m$ .

Input

Input consists of several cases, each one with $m$ followed by the $m + 1$ dimensions. Assume $2 \le m \le 100$ and $1 \le n_i \le 10^4$ .

Output

For every case, print the minimum cost to compute the product of the $m$ matrices.

Problem information

Author: Unknown
Translator: Salvador Roura

Generation: 2026-01-25T09:58:30.977Z