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 Python3 (judge verdict: Accepted)

import sys, os, math, bisect, itertools, collections, heapq, queue, copy, array

# from scipy.sparse.csgraph import csgraph_from_dense, floyd_warshall
# from decimal import Decimal
# from collections import defaultdict, deque

sys.setrecursionlimit(10000000)

ii = lambda: int(sys.stdin.buffer.readline().rstrip())
il = lambda: list(map(int, sys.stdin.buffer.readline().split()))
fl = lambda: list(map(float, sys.stdin.buffer.readline().split()))
iln = lambda n: [int(sys.stdin.buffer.readline().rstrip()) for _ in range(n)]

iss = lambda: sys.stdin.buffer.readline().decode().rstrip()
sl = lambda: list(map(str, sys.stdin.buffer.readline().decode().split()))
isn = lambda n: [sys.stdin.buffer.readline().decode().rstrip() for _ in range(n)]

lcm = lambda x, y: (x * y) // math.gcd(x, y)

MOD = 10 ** 9 + 7
MAX = float('inf')


def main():
    if os.getenv("LOCAL"):
        sys.stdin = open("input.txt", "r")

    N = ii()
    dp = [[0] * (N + 1) for _ in range(N + 1)]
    p = [0] * (N + 1)
    for i in range(N):
        p[i], p[i + 1] = il()

    for l in range(2, N + 1):
        for i in range(1, N - l + 2):
            j = i + l - 1
            dp[i][j] = MAX
            for k in range(i, j):
                dp[i][j] = min(dp[i][j], dp[i][k] + dp[k + 1][j] + p[i - 1] * p[k] * p[j])
    print(dp[1][N])


if __name__ == '__main__':
    main()