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68import sys
"""
Dynamic Programming
Implementation of Matrix Chain Multiplication
Time Complexity: O(n^3)
Space Complexity: O(n^2)
Reference: https://en.wikipedia.org/wiki/Matrix_chain_multiplication
"""
def matrix_chain_order(array: list[int]) -> tuple[list[list[int]], list[list[int]]]:
"""
>>> matrix_chain_order([10, 30, 5])
([[0, 0, 0], [0, 0, 1500], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 0, 0]])
"""
n = len(array)
matrix = [[0 for _ in range(n)] for _ in range(n)]
sol = [[0 for _ in range(n)] for _ in range(n)]
for chain_length in range(2, n):
for a in range(1, n - chain_length + 1):
b = a + chain_length - 1
matrix[a][b] = sys.maxsize
for c in range(a, b):
cost = (
matrix[a][c] + matrix[c + 1][b] + array[a - 1] * array[c] * array[b]
)
if cost < matrix[a][b]:
matrix[a][b] = cost
sol[a][b] = c
return matrix, sol
def print_optimal_solution(optimal_solution: list[list[int]], i: int, j: int):
"""
Print order of matrix with Ai as Matrix.
"""
if i == j:
print("A" + str(i), end=" ")
else:
print("(", end=" ")
print_optimal_solution(optimal_solution, i, optimal_solution[i][j])
print_optimal_solution(optimal_solution, optimal_solution[i][j] + 1, j)
print(")", end=" ")
def main():
"""
Size of matrix created from array [30, 35, 15, 5, 10, 20, 25] will be:
30*35 35*15 15*5 5*10 10*20 20*25
"""
array = [30, 35, 15, 5, 10, 20, 25]
n = len(array)
matrix, optimal_solution = matrix_chain_order(array)
print("No. of Operation required: " + str(matrix[1][n - 1]))
print_optimal_solution(optimal_solution, 1, n - 1)
if __name__ == "__main__":
main()