Permutation and combination is a commonly used concept in mathematics, which is used to represent the way in which several elements are selected from a given set of elements to be arranged or combined. In Python, we can use recursion and iteration to implement efficient permutation and combination algorithms. Below I will show you the code implementation of these two methods.
recursive method
Recursion is a way to solve a problem by having a function call itself over and over again. In permutation and combination algorithms, we can use recursion to generate all possible permutations or combinations.
permutation algorithm
Arrangement is a way of selecting several elements from a given set of elements to arrange. Suppose we have a set containing n elements and want to select r elements for arrangement. We can implement the recursive arrangement algorithm according to the following steps:
- Define a recursive function that accepts three parameters: the list of permutations currently being generated, the list of optional elements, and the number r of elements to be selected.
- If r equals 0, it means that enough elements have been selected and the currently generated permutation is added to the result list.
- Otherwise, for each element in the list of optional elements, do the following in turn:
- Adds this element to the currently generated arrangement.
- Removes this element from the list of optional elements.
- Call the arrangement function recursively, passing in the updated arrangement, the list of optional elements, and r-1.
- Add the element back to the list of selectable elements for the next iteration.
The following is the Python code that uses the recursive method to implement the permutation algorithm:
def permute_recursive(arr, r