Developing a Recursive Program: Listing Permutations

Step #1: Goal and General Algorithm Idea

Scenario: A number of people each need a unique PIN of length n, made up of the digits 1 … n.

Goal: Write a program that generates all possible PINs of length n, made up of the digits 1 … n.

Subgoal: Write a program to generate all permutations of the digits 1 … n.

Let’s begin by looking at the permutations of the digits 1, 2, and 3:

1 2 32 1 3 3 1 2
1 3 22 3 1 3 2 1

Notice a pattern here: pick the first digit 1, permute the other two, and prepend the 1; then pick the second digit 2, permute the other two, and prepend the 2; and finally, pick the third digit 3, permute the other two, and prepend the 3. More generally, we pick the ith digit, permute all the others, and then prepend that ith digit.

This algorithm suggests recursion. It has a base case, where the recursion stops. Specifically, the permutation of 0 digits is empty, and the permutation of 1 digit is that digit itself. And it has an induction step, namely permuting all but the ith digit and then prepending that.

Now that we have the general idea, let’s design the program.

Step #2: Data Representation and Program Structure

Part #1: Data Structures:

Represent the sequence of digits as a list; so the sequence 1, 2, 3 would be treated as a list L.
Represent each permutation as an element of another list I.

Part #2: Functions

And now we write the function suggested by the above. Let’s call it:

function perm(L) → returns list of permutations of elements of L

First, the base case, when there is no recursion and a value is simply returned. This should happen when the list L contains exactly 1 element. We can also add an error check. L should never be the empty list, but we can easily check, and so we do:

if length of L is 0:
  return empty list
if length of L is 1:
  return list containing L

Next, we have to create the list I for the list of permutations. Initially, it’s empty:

I is empty

Now for the recursion. We want to loop through L, extracting the elements successively. After each extraction, we create a new list without it but with all the other elements. We then permute that list, prepend the extracted element, and continue until we are done with the list:

for each element in L:
  remove that element (call it L[i])
  rest of list is L[0 up to i] + L[everything after i]; call this R

  for each element in perm(R):
  prepend L[i]; call the result P
  append P to I

Now we have the list of permutations in I. So we return it.

return I

And that’s it!

Step #3: Put It into Python

We can translate the function above almost line for line:

def perm(L):

# base cases: if list is empty or # has 1 element, return it as a list # so it can be appended to the list # of permutations if len(L) == 0: return [ ] if len(L) == 1: return [ L ]

# this will hold the permuted lists of L I = [ ]

# move each element in the list to the front # and permute the rest of the list; for each # permutation, prepend the front element and # save the result in the list of permutations for i in range(len(L)):

# drop the i-th element; this gives you # the rest of the list to be permuted R = L[:i] + L[i+1:]

# generate the permutations of the rest # for each permutation, prepend the one # you held back and add it to the list of # permutations for e in perm(R): P = [ L[i] ] + e I.append(P)

# return the list of permutations return I

Step #4: The Program

To print the permutations, we just print all the elements in the list that perm returns:

# this is the data to permute
# here, it's numbers, but it can be anything:
data = [ 1, 2, 3, 4 ]

# get the list of permutations and print it for p in perm(data): print(p)

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Matt Bishop
Office: 2209 Watershed Sciences
Phone: +1 (530) 752-8060
ECS 235A, Computer and Information Security
Version of November 5, 2021 at 11:06PM

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