Problem:
MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\\(","\\)"]], processEscapes: true }});

Matrix-chain Multiplication

The goal of the matrix-chain multiplication problem is to find the most efficient way to multiply given $n$ matrices $M_1, M_2, M_3,...,M_n$.

Write a program which reads dimensions of $M_i$, and finds the minimum number of scalar multiplications to compute the maxrix-chain multiplication $M_1M_2...M_n$.

Input

In the first line, an integer $n$ is given. In the following $n$ lines, the dimension of matrix $M_i$ ($i = 1...n$) is given by two integers $r$ and $c$ which respectively represents the number of rows and columns of $M_i$.

Output

Print the minimum number of scalar multiplication in a line.

Constraints

$1 \leq n \leq 100$

$1 \leq r, c \leq 100$

Sample Input 1

6
30 35
35 15
15 5
5 10
10 20
20 25

Sample Output 1

15125

Solution in C# (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace ConsoleApplication1
{
    class Program
    {
        static int[,] dp;
        static int[] p;

        static void Main(string[] args)
        {
            int n = int.Parse(Console.ReadLine());

            dp = new int[n + 1, n + 1];
            p = new int[n + 1];

            string[] input = Console.ReadLine().Split(' ');
            p[0] = int.Parse(input[0]);
            p[1] = int.Parse(input[1]);

            for (int i = 2; i <= n; i++)
            {
                input = Console.ReadLine().Split(' ');
                p[i] = int.Parse(input[1]);
            }

            for (int i = 1; i < n; i++)
            {
                dp[i, i + 1] = p[i - 1] * p[i] * p[i + 1];
            }

            for (int j = 0; j <= n - 3; j++)
            {
                for (int i = 3 + j; i <= n; i++)
                {
                    dp[i - 2 - j, i] = Rec(i - 2 - j, i);
                }
            }

            Console.WriteLine(dp[1, n]);
        }

        static int Rec(int r, int c)
        {
            int min = int.MaxValue;
            int num = c - r;

            for (int i = 0; i < num; i++)
            {
                min = Math.Min(
                    min,
                    dp[r, r + i] + dp[r + i + 1, c] + p[r - 1] * p[r + i] * p[c]);
            }

            return min;
        }
    }
}