"""Factorial series:

1!, 2!, 3!, ...

- Factorial(1) = 1! = 1
- Factorial(2) = 2! = 2
- Factorial(n) = n! = n * (n - 1)! = n * Factorial(n - 1)
"""

from __future__ import absolute_import
from __future__ import print_function
from __future__ import division

def factorial_recur(n):
    """Get the nth number of factorial series, Fn, by recursion.

    - Time complexity: Fn - 1 = O(Fn); too fast.
    - Space complexity: O(n).
    """
    if n <= 1:
        return 1
    else:
        return n * factorial_recur(n - 1)


def factorial_memo(n):
    """Get the nth number of factorial series, Fn, by memorization.
    
    - Time complexity: O(n).
    - Space complexity: O(n).
    """
    fn_d = {}
    fn_d[0] = 1
    fn_d[1] = 1
    for n in xrange(2, n + 1):
        fn_d[n] = n * fn_d[n - 1]
    return fn_d[n]

def factorial_dp(n):
    """Get the nth number of factorial series by dynamic programming.

    - Time complexity is still O(n), like factorial_memo().
    - Space complexity is O(1), improving a lot.
    """
    fn = 1
    for i in xrange(2, n + 1):
        fn *= i 
    return fn


def main():
    import time
    n = 10
    
    print('{}th number of factorial series:'.format(n))

    start_time = time.time()
    print('By recursion: {}'.format(factorial_recur(n)))
    print('Time: {}'.format(time.time() - start_time))

    start_time = time.time()
    print('By memorization: {}'.format(factorial_memo(n)))
    print('Time: {}'.format(time.time() - start_time))

    start_time = time.time()
    print('By dynamic programming: {}'.format(factorial_dp(n)))
    print('Time: {}'.format(time.time() - start_time))

if __name__ == '__main__':
    main()

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