Python | All possible permutations of N lists

Computing permutations is always a necessary task in many of the practical applications and a concept widely used in Mathematics to achieve solutions to many practical problems. Lets discuss certain ways in which one can perform the task of getting all the permutations of N lists. 

Method #1 : Using list comprehension List comprehension can be used to convert the naive method task into a single line, hence more compact. This method checks for each element available elements and makes pairs accordingly. 

Python3

# Python3 code to demonstrate
# to compute all possible permutations
# using list comprehension

# initializing lists
list1 = [1, 3, 4]
list2 = [6, 7, 9]
list3 = [8, 10, 5]

# printing lists
print ("The original lists are : " + str(list1) +
                            " " + str(list2) +
                            " " + str(list3))

# using list comprehension
# to compute all possible permutations
res = [[i, j, k] for i in list1
                for j in list2
                for k in list3]

# printing result
print ("All possible permutations are : " + str(res))

Output

The original lists are : [1, 3, 4] [6, 7, 9] [8, 10, 5]
All possible permutations are : [[1, 6, 8], [1, 6, 10], [1, 6, 5], [1, 7, 8], [1, 7, 10], [1, 7, 5], [1, 9, 8], [1, 9, 10], [1, 9, 5], [3, 6, 8], [3, 6, 10], [3, 6, 5], [3, 7, 8], [3, 7, 10], [3, 7, 5], [3, 9, 8], [3, 9, 10], [3, 9, 5], [4, 6, 8], [4, 6, 10], [4, 6, 5], [4, 7, 8], [4, 7, 10], [4, 7, 5], [4, 9, 8], [4, 9, 10], [4, 9, 5]]

Time Complexity: O(n³) where n is the length of the list.
Space Complexity: O(n³) where n is the length of the list.

Method #2 : Using itertools.product() Using product function, one can easily perform this task in more pythonic and concise manner. This is most recommended method to perform this task of computing cartesian product. 

Python3

# Python3 code to demonstrate
# to compute all possible permutations
# using itertools.product()
import itertools

# initializing list of list
all_list = [[1, 3, 4], [6, 7, 9], [8, 10, 5] ]

# printing lists
print ("The original lists are : " + str(all_list))

# using itertools.product()
# to compute all possible permutations
res = list(itertools.product(*all_list))

# printing result
print ("All possible permutations are : " + str(res))

Output

The original lists are : [[1, 3, 4], [6, 7, 9], [8, 10, 5]]
All possible permutations are : [(1, 6, 8), (1, 6, 10), (1, 6, 5), (1, 7, 8), (1, 7, 10), (1, 7, 5), (1, 9, 8), (1, 9, 10), (1, 9, 5), (3, 6, 8), (3, 6, 10), (3, 6, 5), (3, 7, 8), (3, 7, 10), (3, 7, 5), (3, 9, 8), (3, 9, 10), (3, 9, 5), (4, 6, 8), (4, 6, 10), (4, 6, 5), (4, 7, 8), (4, 7, 10), (4, 7, 5), (4, 9, 8), (4, 9, 10), (4, 9, 5)]

Time Complexity: O(n^k) where n is the number of lists and k is the number of elements of each list.
Auxiliary Space: O(n^k) where n is the number of lists and k is the number of elements of each list.

Method #3: Using numpy

In this method, numpy library is used to create a 3D array of all possible combinations of elements from the three lists using the np.meshgrid() function. The resulting 3D array is reshaped into a 2D array and stored in the res variable.

Python3

import numpy as np

# initializing lists as arrays
list1 = np.array([1, 3, 4])
list2 = np.array([6, 7, 9])
list3 = np.array([8, 10, 5])

# printing arrays
print("The original arrays are:")
print(list1)
print(list2)
print(list3)

# using NumPy meshgrid and stack
# to compute all possible permutations
X, Y, Z = np.meshgrid(list1, list2, list3)
res = np.column_stack((X.ravel(), Y.ravel(), Z.ravel()))

# printing result
print("All possible permutations are:")
print(res)

# Contributed by rishabmalhdijo

Output: 

Output

Time Complexity: O(n³) where n is the length of each list
Space Complexity: O(n³), because the resulting 3D array has to be stored in memory.