Problem:
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 (Python):



import sys


def mat_mul_opt(_mat_info):
    rc_info = list(map(int, _mat_info[0]))
    rc_info.extend(list(map(lambda x: int(x[1]), _mat_info[1:])))

    for l in range(2, mat_num + 1):
        for i in range(1, mat_num - l + 2):
            j = i + l - 1
            ans[i][j] = 1e7
            for k in range(i, j):
                ans[i][j] = min(ans[i][j], ans[i][k] + ans[k + 1][j] +
                                rc_info[i - 1] * rc_info[k] * rc_info[j])

    return ans[1][mat_num]


if __name__ == '__main__':
    _input = sys.stdin.readlines()
    mat_num = int(_input[0])
    mat_info = tuple(map(lambda x: x.split(), _input[1:]))
    

    ans = [[0] * (mat_num + 1) for _ in range(mat_num + 1)]
    print(mat_mul_opt(mat_info))

Solution (Python):



import sys


def mat_mul_opt(_mat_info):
    rc_info = list(map(int, _mat_info[0]))
    rc_info.extend(list(map(lambda x: int(x[1]), _mat_info[1:])))

    ans = [[0] * (mat_num + 1) for _ in range(mat_num + 1)]

    for l in range(2, mat_num + 1):
        for i in range(1, mat_num - l + 2):
            j = i + l - 1
            ans[i][j] = 1e7
            for k in range(i, j):
                ans[i][j] = min(ans[i][j], ans[i][k] + ans[k + 1][j] +
                                rc_info[i - 1] * rc_info[k] * rc_info[j])

    return ans[1][mat_num]


if __name__ == '__main__':
    _input = sys.stdin.readlines()
    mat_num = int(_input[0])
    mat_info = tuple(map(lambda x: x.split(), _input[1:]))
    

    print(mat_mul_opt(mat_info))

Solution (Python):



import sys


def mat_mul_opt(_mat_info):
    rc_info = list(map(int, _mat_info[0]))
    rc_info.extend(list(map(lambda x: int(x[1]), _mat_info[1:])))

    ans = [[0] * (mat_num + 1) for _ in range(mat_num + 1)]

    for l in range(1, mat_num):
        for i in range(1, mat_num - l + 1):
            j = i + l
            ans[i][j] = 1e7
            for k in range(i, j):
                ans[i][j] = min(ans[i][j], ans[i][k] + ans[k + 1][j] +
                                rc_info[i - 1] * rc_info[k] * rc_info[j])

    return ans[1][mat_num]


if __name__ == '__main__':
    _input = sys.stdin.readlines()
    mat_num = int(_input[0])
    mat_info = tuple(map(lambda x: x.split(), _input[1:]))
    

    print(mat_mul_opt(mat_info))

Solution (Python):



import sys


def mat_mul_opt(_mat_info):
    rc_info = list(map(int, _mat_info[0]))
    rc_info.extend(list(map(lambda x: int(x[1]), _mat_info[1:])))

    ans = [[0] * (mat_num + 1) for _ in range(mat_num + 1)]

    for l in range(2, mat_num + 1):
        for i in range(1, mat_num - l + 2):
            j = i + l - 1
            ans[i][j] = float('inf')
            for k in range(i, j):
                ans[i][j] = min(ans[i][j], ans[i][k] + ans[k + 1][j] +
                                rc_info[i - 1] * rc_info[k] * rc_info[j])

    return ans[1][mat_num]


if __name__ == '__main__':
    _input = sys.stdin.readlines()
    mat_num = int(_input[0])
    mat_info = tuple(map(lambda x: x.split(), _input[1:]))
    

    print(mat_mul_opt(mat_info))

Solution (Python):
def matrixChainMultiplication(p):
    
    n = len(p) - 1
    
    
    cost = [[0 for _ in range(n + 1)] for _ in range(n + 1)]
    for numOfMatrices in range(2, n + 1):
        for i in range(1, n - numOfMatrices + 2):
            j = i + numOfMatrices - 1
            cost[i][j] = float('inf')
            for k in range(i, j):
                cost[i][j] = min(cost[i][j], cost[i][k] + cost[k + 1][j] + p[i - 1] * p[k] * p[j])
    print(cost[1][n])

p = []
n = int(input())

for _ in range(n):
    listed_input = input().split()
    p.append(int(listed_input[0]))
p.append(int(listed_input[1]))

matrixChainMultiplication(p)

Solution (Python):
n = int(input())

p = [0 for i in range(n+1)]
for i in range(n):
    p[i],p[i+1] = map(int,input().split())
    
m = [[0 for i in range(n)] for j in range(n)]


for l in range(1,n):
    for i in range(0,n-l):
        j = i + l
        m[i][j] = float("inf")
        for k in range(i,j):
            m[i][j] = min(m[i][j],m[i][k] +m[k+1][j] + p[i]*p[k+1]*p[j+1])   
   
print(m[0][n-1])

Solution (Python):
n = int(input())

p = [0 for i in range(n+1)]
for i in range(n):
    p[i],p[i+1] = map(int,input().split())
    
m = [[0 for i in range(n)] for j in range(n)]

for l in range(1,n):
    for i in range(0,n-l):
        j = i + l
        m[i][j]= float("inf")
        for k in range(i,j):
            m[i][j] = min(m[i][j],m[i][k] + m[k+1][j] + p[i]*p[k+1]*p[j+1])


print(m[0][n-1])

Solution (Python):


import random
import sys
import os
import pprint


"""Matrix Chain Multiplication"""




n = int(input())
col = [0] * (n + 1)
for i in range(n):
    row, column = map(int, input().split())
    col[i] = row
    col[i+1] = column

memo = [[0 for i in range(n+1)] for j in range(n+1)]


for length in range(2, n+1):
    for start_index in range(1, n - length + 2):
        end_index = start_index + length - 1
        candidates = []
        for split_index in range(start_index, end_index):
            left = memo[start_index][split_index]
            right = memo[split_index + 1][end_index]
            plus = col[start_index - 1] * col[split_index] * col[end_index]
            candidates.append(left + plus + right)

        memo[start_index][end_index] = min(candidates)
        


print(memo[1][n])

Solution (Python):
n = int(input())
p = [list(map(int, input().split())) for i in range(n)]

m = [[0] * n for i in range(n)]

for l in range(2, n + 1):
  for i in range(n - l + 1):
    j = i + l - 1
    m[i][j] = float('inf')
    for k in range(i, j):
      cost = m[i][k] + m[k + 1][j] + p[i][0] * p[k][1] * p[j][1]
      m[i][j] = min(m[i][j], cost)

print(m[0][n - 1])

Solution (Python):
def maxChain(n,p):
    m = [[0 for i in range(n+1)] for i in range(n+1)]
    for l in range(2, n+1):
        for i in range(1, n - l + 2):
            j = i + l - 1
            m[i][j] = float("inf")
            for k in range(i, j):
                m[i][j] = min(m[i][j], m[i][k] + m[k+1][j] + p[i - 1] * p[k] * p[j])
    return str(m[1][n])
n = int(input())
p = []
for i in range(n):
    L = list(map(int, input().split()))
    if i == 0:
        p.append(L[0])
        p.append(L[1])
    else:
        p.append(L[1])
print(maxChain(n,p))

Solution (Python):
def matrixChainMultiplication(M):
    
    n = len(M)
    P = [M[0][0]]
    P += [M[i][1] for i in range(n)]
    
    m = [[[] for _ in range(n + 1)] for _ in range(n + 1)]
    for i in range(n + 1):
            m[i][i] = 0

    
    for l in range(2, n + 1):
        for i in range(1, n - l + 2):
            j = i + l - 1
            m[i][j] = 9999999999999999
            for k in range(i, j):
                m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + P[i -1] * P[k] * P[j])
    return m[1][n]


if __name__ == '__main__':
    n = int(input())
    M = [list(map(int, input().split())) for _ in range(n)]
    print(matrixChainMultiplication(M))

Solution (Python):
def matrixChainMultiplication(M):
    
    n = len(M)
    P = [M[0][0]]
    P += [M[i][1] for i in range(n)]
    
    m = [[[] for _ in range(n + 1)] for _ in range(n + 1)]
    for i in range(n + 1):
            m[i][i] = 0

    
    for l in range(2, n + 1):
        for i in range(1, n - l + 2):
            j = i + l - 1
            m[i][j] = 9999999999999999
            for k in range(i, j):
                m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + P[i -1] * P[k] * P[j])
    return m[1][n]


if __name__ == '__main__':
    n = int(input())
    M = [list(map(int, input().split())) for _ in range(n)]
    print(matrixChainMultiplication(M))

Solution (Python):
import sys

sys.setrecursionlimit(10**7)

inf = float('inf')

def solve():
    n = int(sys.stdin.readline())

    r = [0] * n
    c = [0] * n

    for i in range(n):
        ri, ci = map(int, sys.stdin.readline().split())
        r[i] = ri
        c[i] = ci

    memo = [[-1]*(n + 1) for i in range(n + 1)]

    ans = mcm(memo, n, r, c, 0, n)

    

    print(ans)

def mcm(memo, n, r, c, left, right):
    if memo[left][right] != -1:
        return memo[left][right]
    elif right - left == 1:
        memo[left][right] = 0
        return 0
    else:
        res = inf

        for i in range(left + 1, right):
            res = min(res, mcm(memo, n, r, c, left, i) + mcm(memo, n, r, c, i, right) + r[left]*r[i]*c[right-1])

        memo[left][right] = res
        return res

def compute_lcs(x, y):
    n = len(x)
    m = len(y)

    x = '
    y = '

    dp = [[0]*(m + 1) for i in range(n + 1)]

    for i in range(1, n + 1):
        for j in range(1, m + 1):
            if x[i] == y[j]:
                dp[i][j] = dp[i - 1][j - 1] + 1
            else:
                dp[i][j] = max(dp[i - 1][j], dp[i][j - 1])

    return dp[n][m]

if __name__ == '__main__':
    solve()

Solution (Python):
import sys

sys.setrecursionlimit(10**7)

inf = float('inf')

def solve():
    n = int(sys.stdin.readline())

    r = [0] * n
    c = [0] * n

    for i in range(n):
        ri, ci = map(int, sys.stdin.readline().split())
        r[i] = ri
        c[i] = ci

    dp = [[0]*(n + 1) for i in range(n + 1)]

    for ls in range(2, n + 1):
        for i in range(n - ls + 1):
            dp[i][i + ls] = inf
            for j in range(i + 1, i + ls):
                dp[i][i + ls] = min(dp[i][i + ls], dp[i][j] + dp[j][i + ls] + r[i]*r[j]*c[i + ls - 1])

    ans = dp[0][n]

    print(ans)

if __name__ == '__main__':
    solve()

Solution (Python):
n = int(input())
m = [[0 for j in range(n + 1)] for i in range(n + 1)]
z = [0] * (n + 1)
for i in range(1, n + 1):
    z[i - 1], z[i] = map(int, input().split())
j = 0
for l in range(2, n + 1):
    for i in range(1, n - l + 2):
        j = i + l - 1
        m[i][j] = int(1e8)
        for k in range(i, j):
            m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + z[i - 1] * z[k] * z[j])
print(m[1][n])

Solution (Python):
n = int(input())
m = [[0] * (n + 1) for _ in range(n + 1)]
z = [0] * (n + 1)
for i in range(n):
    z[i], z[i + 1] = map(int, input().split())
j = 0
for l in range(2, n + 1):
    for i in range(1, n - l + 2):
        j = i + l - 1
        m[i][j] = float("inf")
        for k in range(i, j):
            m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + z[i - 1] * z[k] * z[j])
print(m[1][n])

Solution (Python):



import sys
import re
import math
import collections
import itertools
import functools
import logging

logging.basicConfig(level=logging.DEBUG, format="DBG: %(message)s")

def dbg(msg, *args, **kwargs):
    logging.debug(msg, *args, **kwargs)

Mat = collections.namedtuple("Mat", "r c")

def mat_mul(ms):
    return Mat(ms[0].r, ms[-1].c)

def cost(m1, m2):
    return m1.r * m1.c * m2.c

def solve(n, ms):
    if n == 1: return 0

    dp = {}
    for i in range(n):
        dp[(i,i)] = 0

    for l in range(2, n+1):
        for i in range(n-l+1):
            
            cost_min = float("inf")
            for j in range(l-1):
                
                m_lhs = mat_mul(ms[i:i+j+1])
                m_rhs = mat_mul(ms[i+j+1:i+l])
                
                cost_lhs = dp[(i,i+j)]
                cost_rhs = dp[(i+j+1,i+l-1)]
                
                cost_mul = cost(m_lhs, m_rhs)
                
                cost_now = cost_mul + cost_lhs + cost_rhs
                
                if cost_min > cost_now:
                    cost_min = cost_now
            dp[(i,i+l-1)] = cost_min

    
    return dp[(0,n-1)]

def main():
    n = int(input())

    ms = []
    for _ in range(n):
        r,c = map(int, input().split())
        ms.append(Mat(r,c))

    print(solve(n, ms))

if __name__ == "__main__": main()

Solution (Python):



import sys
import re
import math
import collections
import itertools
import functools

DEBUG = True
DEBUG = False

def dbg(*args):
    if DEBUG:
        print("DBG: ", file=sys.stderr, end="")
        print(*args, file=sys.stderr)

Mat = collections.namedtuple("Mat", "r c")

def mat_mul(ms):
    return Mat(ms[0].r, ms[-1].c)

def cost(m1, m2):
    return m1.r * m1.c * m2.c

def solve(n, ms):
    if n == 1: return 0

    dp = {}
    for i in range(n):
        dp[(i,i)] = 0

    for l in range(2, n+1):
        for i in range(n-l+1):
            
            cost_min = float("inf")
            for j in range(l-1):
                dbg("l, i, j: %d %d %d", l, i, j)
                m_lhs = mat_mul(ms[i:i+j+1])
                m_rhs = mat_mul(ms[i+j+1:i+l])
                dbg("m_lhs, m_rhs: %r %r", m_lhs, m_rhs)
                cost_lhs = dp[(i,i+j)]
                cost_rhs = dp[(i+j+1,i+l-1)]
                dbg("cost_lhs, cost_rhs: %d %d", cost_lhs, cost_rhs)
                cost_mul = cost(m_lhs, m_rhs)
                dbg("cost_mul: %d", cost_mul)
                cost_now = cost_mul + cost_lhs + cost_rhs
                dbg("cost_now: %d", cost_now)
                if cost_min > cost_now:
                    cost_min = cost_now
            dp[(i,i+l-1)] = cost_min

    dbg(dp)
    return dp[(0,n-1)]

def main():
    n = int(input())

    ms = []
    for _ in range(n):
        r,c = map(int, input().split())
        ms.append(Mat(r,c))

    print(solve(n, ms))

if __name__ == "__main__": main()

Solution (Python):
def matrixChainMultiplication():
    for i in range(1, n+1):
        m[i][i] = 0

    for l in range(2, n+1):
        for i in range(1, n-l+2):
            j = i + l - 1
            m[i][j] = INF
            for k in range(i, j):
                m[i][j] = min(m[i][j], m[i][k] + m[k+1][j] + p[i-1] * p[k] * p[j])
    return m[1][n]

if __name__ == "__main__":
    INF = 10000000000000
    p = []
    n = int(input())
    m = [[None] * (n+1) for i in range(n+1)]
    for i in range(n):
        inp = list(map(int, input().split()))
        if i == 0:
            p.append(inp[0])
            p.append(inp[1])
        else:
            p.append(inp[1])
    print(matrixChainMultiplication())

Solution (Python):
from pprint import pprint

n = int(input().strip())
for i in range(n):
    line = input().strip().split()
    if i == 0:
        mat = [int(line[0]), int(line[1])]
    else:
        mat.append(int(line[1]))

dp = [[-999999999] * (n+1) for _ in range(n+1)]
for i in range(n+1):
    dp[i][i] = 0

for i in range(n, 0, -1):
    for j in range(i+1, n+1):
        
        
        
        
        
        
        dp[i][j] = min(dp[i][k] + dp[k+1][j] + mat[i-1] * mat[k] * mat[j] for k in range(i, j))
        


print(dp[1][n])

Solution (Python):



def calc_cost():
    global M
    for diff in range(0, n):
        for i in range(1, n-diff+1):
            j = i + diff
            if diff == 0:
                M[i][j] = 0
            elif diff == 1:
                M[i][j] = p[i-1] * p[i] * p[j]
            else:
                M[i][j] = calc_min(i, j)
            
            
            
    return M[1][n]

def calc_min(i, j):
    global M, p
    min_cost = float("inf")
    for k in range(i, j):
        cost = M[i][k] + M[k+1][j] + (p[i-1] * p[k] * p[j])
        
        if cost < min_cost:
            min_cost = cost
    return min_cost

n = int(input())
M = [["-" for _ in range(n+1)] for _ in range(n+1)]
r_list = []
c_list = []
p = []
for _ in range(n):
    r, c = map(int, input().split())
    r_list.append(r)
    c_list.append(c)

p.append(r_list[0])
for c in c_list:
    p.append(c)

total_cost = calc_cost()
print(total_cost)

Solution (Python):
def main():
    n = input()
    p = map(int,raw_input().split())
    for i in xrange(1,n):
        p.append(map(int,raw_input().split())[1])

    print mcm(p)

def mcm(p):
    n = len(p)-1
    m = [[0]*(n+1) for i in xrange(n+1)]

    for l in xrange(2,n+1):
        for i in xrange(1,n-l+2):
            j = i+l-1
            m[i][j] = float("inf")
            for k in xrange(i,j):
                m[i][j] = min(m[i][j],m[i][k]+m[k+1][j]+p[i-1]*p[k]*p[j])

    return m[1][n]
            

if __name__ == "__main__":
    main()

Solution (Python):
n = int(raw_input())
matrix = []
for i in range(n):
    a = map(int, raw_input().split())
    matrix.append([a[0], a[1]])

m = [[0 for i in range(n+1)] for j in range(n+1)]
p = [0 for i in range(n+1)]
p[0] = matrix[0][0]
for i in range(n):
    p[i+1] = matrix[i][1]

for l in range(2, n+1):
    for i in range(1, n-l+2):
        j = i + l - 1
        m[i][j] = 10**19
        for k in range(i, j):
            m[i][j] = min(m[i][j], m[i][k] + m[k+1][j] + p[i-1] * p[k] * p[j])

print m[1][n]

Solution (Python):
INF = 1e9

def MatrixChainMulti(n, p):
    dp = [[0 for _ in range(n+1)] for _ in range(n+1)]
    
    for i in range(n+1):
        dp[i][i] = 0
        
    for l in range(1, n): 
        for i in range(0, n-l+1): 
            j = i + l
            dp[i][j] = INF
            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])
    
    return dp[1][n]

N = int(input())
p = []
a, b = list(map(int, input().split()))
p.append(a)
p.append(b)

for _ in range(N-1):
    c, d = list(map(int, input().split()))
    p.append(d)
    
print(MatrixChainMulti(N, p))

Solution (Python):

N = int(input())
p = [0 for _ in range(N+1)]

p[0], p[1] = map(int, input().split())
for n in range(2, N+1):
    p[n] = list(map(int, input().split()))[1]

M = [[0]*(N+1) for _ in range(N+1)]

for n in range(N+1):
    M[n][n] = 0

for l in range(2, N+1):
    for i in range(1, N-l+2):
        j = i + l - 1
        M[i][j] = float("inf")
        for k in range(i, j):
            M[i][j] = min(M[i][j], M[i][k] + M[k+1][j] + p[i-1]*p[k]*p[j])

print(M[1][N])

Solution (Python):
def matrix_chain_multiplication(n, list):
    matrix = [[0 for i in range(n)] for j in range(n)]

    for i in range(n-1):
        for j in range(n-i-1):
            k = j+i+1
            matrix[j][k] = float('inf')
            for r in range(j, k):
                matrix[j][k] = min(matrix[j][k],
                    matrix[j][r] + matrix[r+1][k]
                    + list[j]*list[r+1]*list[k+1])

    return matrix[0][n-1]

if __name__ == '__main__':
    n = int(input())

    list = list(map(int, input().split()))
    for i in range(n-1):
        list.append(int(input().split()[1]))

    print(matrix_chain_multiplication(n, list))

Solution (Python):
def main():
    """ MAIN """
    DAT = []
    num = int(input())
    for i in range(num):
        x, y = list(map(int,input().split()))
        DAT.append(x)
    DAT.append(y)
    num1 = num + 1
    TBL = [[0] * num1 for i in range(num1)]

    for i in range(0, num1):
        for row in range(num-i):
            col = row + i + 1
            for j in range(row+1,col):
                x = DAT[row] * DAT[j] * DAT[col] + TBL[row][j] + TBL[j][col]
                if TBL[row][col] < 1 or TBL[row][col] > x:
                    TBL[row][col] = x

    print("{}".format(TBL[row][col]))

if __name__ == '__main__':
    main()

Solution (Python):
if __name__ == '__main__':
    N = input()


    m = [[0] * (N+1) for i in range(N+1)]

    p = [0] * N

    for i in range (N):
        p[i],dummy  = map(int, raw_input().split())
        
    p.append(dummy)

    for l in range (2, N+1):
        for i in range(1, N - l + 2):
            j = i + l - 1
            m[i][j] = float("inf")
            for k in range(i, j):
                m[i][j] = min(m[i][j], m[i][k] + m[k+1][j] + p[i-1] * p[k] * p[j])

print  m[1][N]

Solution (Python):
n = int(input())
p = list(range(n+2))

for i in range(1,n+1):
    p[i], p[n+1] = map(int, input().split())


table = [[0] * (n+1) for i in range(n+1)]


for i in range(1,n):  
    table[i][i+1] = p[i]*p[i+1]*p[i+2]


for k in range(3, n+1):    
    for i in range(1,n+2-k): 
        j = i + k - 1      

        table[i][j] = float('inf')

        for x in range(i, j): 
            
            cost = table[i][x] + table[x+1][j] + p[i]*p[x+1]*p[j+1]
            table[i][j] = min(table[i][j], cost)
            

print(table[1][n])

Solution (Python):
n = int(input())
INF = 10**9
P = [list(map(int, input().split())) for i in range(n)]
m = [[0]*n for i in range(n)]

for l in range(2, n + 1):
    for i in range(n - l + 1):
        j = i + l - 1
        m[i][j] = INF
        for k in range(i, j):
            m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + P[i][0] * P[k][1] * P[j][1])

print(m[0][n - 1])

Solution (Python):
n = int(input())
p = [list(map(int, input().split())) for i in range(n)]

m = [[0] * n for i in range(n)]

for l in range(2, n + 1):
    for i in range(n - l + 1):
        j = i + l - 1
        m[i][j] = float('inf')
        for k in range(i, j):
            m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + p[i][0] * p[k][1] * p[j][1])

print(m[0][n - 1])

Solution (Python):


n = int(input())
p = []
m = dict()
for _ in range(n):
    r, c = map(int, input().split())
    p.append(r)
    if _ == n-1:
        p.append(c)

for i in range(n+1):
    m[(i,i)] = 0

for l in range(2, n + 1):
    for i in range(1, n - l + 2):
        j = i + l - 1
        m[(i, j)] = float('inf')
        
        for k in range(i, j):
            value = m[(i, k)] + m[(k+1, j)] + p[i-1]*p[k]*p[j]
            m[(i, j)] = min(m[(i, j)], value)

print(m[(1, n)])

Solution (Python):



if __name__ == '__main__':

    n = int(input())
    p = [0]
    for _ in range(n):
        r, c = list(map(int, input().split(" ")))
        p[-1] = r
        p.append(c)
    m = [[0 for i in range(n+1)] for j in range(n+1)]
    
    for i in range(1, n+1):
        m[i][i] = 0
    
    for l in range(2, n+1):
        for i in range(1, n-l+2):
            j = i + l - 1
            m[i][j] = float("inf")
            for k in range(i, j):
                m[i][j] = min(m[i][j], m[i][k] + m[k+1][j] + p[i-1]*p[k]*p[j])

    print(m[1][n])

Solution (Python):
import sys
INF = float("inf")
n = int(input())
A = [0]
for i, a in enumerate(sys.stdin):
    a = list(map(int, a.split()))
    if i == 0:
        A.extend(a)
    else:
        A.append(a[1])
C = [[INF] * (n + 1) for _ in range(n + 1)]
for i in range(n + 1):
    C[i][i] = 0
for j in range(1, n):
    for i in range(1, n - j + 1):
        for k in range(i + 1, i + j + 1):
            cost = C[i][k - 1] + C[k][i + j] + A[i] * A[k] * A[i + j + 1]
            if cost < C[i][i + j]:
                C[i][i + j] = cost
print(C[1][n])

Solution (Python):
INF = float("inf")
MAX = 101
n = int(input())
z = []
m = [[0] * MAX for _ in range(MAX)]
for i in range(1, n + 1):
    a, b = map(int, input().split())
    z.append(a)
z.append(b)
for l in range(2, n + 1):
    for i in range(1, n - l + 2):
        j = i + l - 1
        m[i][j] = INF
        for k in range(i, j):
            m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + z[i - 1] * z[k] * z[j])
print(m[1][n])

Solution (Python):
INF = float("inf")
n = int(input())
z = []
m = [[0] * (n + 1) for _ in range(n + 1)]
for i in range(1, n + 1):
    a, b = map(int, input().split())
    z.append(a)
z.append(b)
for l in range(1, n):
    for i in range(1, n - l + 1):
        j = i + l
        m[i][j] = INF
        for k in range(i, j):
            m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + z[i - 1] * z[k] * z[j])
print(m[1][n])

Solution (Python):
import sys
INF = float("inf")
n = int(input())
m = [[0] * (n + 1) for _ in range(n + 1)]
li = [list(map(int, a.split())) for a in sys.stdin]
z = (li[0][0], ) + list(zip(*li))[1]
for l in range(1, n):
    for i in range(1, n - l + 1):
        j = i + l
        m[i][j] = INF
        for k in range(i, j):
            m[i][j] = min(m[i][j], m[i][k] + m[k + 1][j] + z[i - 1] * z[k] * z[j])
print(m[1][n])

Solution (Python):
n=int(input())+1
e=[input().split()for _  in[0]*~-n]
p=[int(e[0][0])]+list(int(x[1])for x in e)
m=[[0]*n for _ in[0]*n]
for l in range(2,n):
 for i in range(1,n-l+1):
  j=i+l-1;m[i][j]=1e6
  for k in range(i,j):m[i][j]=min(m[i][j],m[i][k]+m[k+1][j]+p[i-1]*p[k]*p[j])
print(m[1][n-1])

Solution (Python):
n=int(input())+1
e=[input().split()for _  in[0]*~-n]
p=[int(e[0][0])]+list(int(x[1])for x in e)
m=[[0]*n for _ in[0]*n]
for l in range(2,n):
 for i in range(1,n-l+1):
  j=i+l-1;m[i][j]=1e6
  for k in range(i,j):m[i][j]=min(m[i][j],m[i][k]+m[k+1][j]+p[i-1]*p[k]*p[j])
print(m[1][n-1])

Solution (Python):
n=int(input())+1
e=[input().split()for _  in[0]*~-n]
p=[int(e[0][0])]+list(int(x[1])for x in e)
m=[[0]*n for _ in[0]*n]
for l in range(2,n):
 for i in range(1,n-l+1):
  j=i+l-1;m[i][j]=1e6
  for k in range(i,j):m[i][j]=min(m[i][j],m[i][k]+m[k+1][j]+p[i-1]*p[k]*p[j])
print(m[1][n-1])

Solution (Python):
import sys
n=int(input())+1
e=list(map(lambda x:x.split(),sys.stdin))
p=[int(e[0][0])]+list(int(x[1])for x in e)
m=[[0]*n for _ in[0]*n]
for l in range(2,n):
 for i in range(1,n-l+1):
  j=i+l-1;m[i][j]=1e6
  for k in range(i,j):m[i][j]=min(m[i][j],m[i][k]+m[k+1][j]+p[i-1]*p[k]*p[j])
print(m[1][n-1])

Solution (Python):
import sys
n=int(input())+1
e=list(map(lambda x:x.split(),sys.stdin))
p=[int(e[0][0])]+list(int(x[1])for x in e)
m=[[0]*n for _ in[0]*n]
for l in range(2,n):
 for i in range(1,n-l+1):
  j=i+l-1;m[i][j]=1e6
  for k in range(i,j):m[i][j]=min(m[i][j],m[i][k]+m[k+1][j]+p[i-1]*p[k]*p[j])
print(m[1][n-1])

Solution (Python):
import sys
n=int(input())+1
e=list(map(lambda x:x.split(),sys.stdin))
p=[int(e[0][0])]+list(int(x[1])for x in e)
m=[[0]*n for _ in[0]*n]
for l in range(2,n):
 for i in range(1,n-l+1):
  j=i+l-1;m[i][j]=1e6
  for k in range(i,j):m[i][j]=min(m[i][j],m[i][k]+m[k+1][j]+p[i-1]*p[k]*p[j])
print(m[1][n-1])

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for l in range(2,n):
  for i in range(1,n-l+1):
   j=i+l-1;m[i][j]=1e6
   for k in range(i,j):m[i][j]=min(m[i][j],m[i][k]+m[k+1][j]+p[i-1]*p[k]*p[j])
 print(m[1][n-1])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
import sys
def s():
 n=int(input())+1
 e=list(map(lambda x:x.split(),sys.stdin))
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
import sys
def s():
 n=int(input())+1
 e=list(map(lambda x:x.split(),sys.stdin))
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
import sys
def s():
 n=int(input())+1
 e=list(map(lambda x:x.split(),sys.stdin))
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if 1>m[r][c]or m[r][c]>x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()

Solution (Python):
def s():
 n=int(input())+1
 e=[input().split()for _  in[0]*~-n]
 p=[int(e[0][0])]+list(int(x[1])for x in e)
 m=[[0]*n for _ in[0]*n]
 for i in range(n):
  for r in range(n-i-1):
   c=r+i+1
   for j in range(r+1,c):
    x=m[r][j]+m[j][c]+p[r]*p[j]*p[c]
    if not 0<m[r][c]<=x:m[r][c]=x
 print(m[r][c])
if'__main__'==__name__:s()