Generating Permutations

1
Generating Permutations

• Bottom Up
• Johnson-Trotter
• Lexicographic

2
Bottom Up Permutations

• Given the (n1)! permutations of 1,,n1,
  insert n into each position of each of them.
• This generates n! permutations of 1,,n.
• Base step 1
• Step 1 1 2, 2 1
• Step 2 1 2 3, 1 3 2, 3 1 2 3 2 1, 2 3 1, 2
  1 3
• Step 3 1234, 1243, 1423, 4123 4132, 1432,
  1342, 1324
• 3124, 3142, 3412, 4312 4321, 3421,
  3241, 3214
• 2314, 2341, 2431, 4231 4213, 2413,
  2143, 2134
• Notice the ordering This is called minimal
  change.

3
Mobile Elements forMinimal Change Algorithm

• If each element has an associated direction, then
  we call those elements mobile that are facing
  smaller adjacent elements.
• ????
• 3 2 4 1
• In this example, 3 4 are mobile, while 1 2
  are not.
• We start out with the increasing permutation,
  with each element facing left.

4
Johnson-Trotter Algorithm

• INPUT a positive integer n
• OUTPUT list of all permutations of 1,,n
• initialize first permutation
• while ? a mobile element do
• find the largest mobile element k
• swap k and the adjacent element it is facing
• reverse the direction of all elements larger than
  k
• print current permutation

5
Example
1 2 3 4

• ????
• 1 2 3 4

6
Example
1 2 3 4 1 2 4 3

• ????
• 1 2 4 3

7
Example
1 2 3 4 1 2 4 3 1 4 2 3

• ????
• 1 4 2 3

8
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3

• ????
• 4 1 2 3

9
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2

• ????
• 4 1 3 2

10
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2

• ????
• 1 4 3 2

11
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2

• ????
• 1 3 4 2

12
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4

• ????
• 1 3 2 4

13
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4

• ????
• 3 1 2 4

14
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2

• ????
• 3 1 4 2

15
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2

• ????
• 3 4 1 2

16
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2

• ????
• 4 3 1 2

17
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1

• ????
• 4 3 2 1

18
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1

• ????
• 3 4 2 1

19
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1

• ????
• 3 2 4 1

20
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4

• ????
• 3 2 1 4

21
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4

• ????
• 2 3 1 4

22
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4 2 3 4 1

• ????
• 2 3 4 1

23
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4 2 3 4 1

• ????
• 2 4 3 1

2 4 3 1
24
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4 2 3 4 1

• ????
• 4 2 3 1

2 4 3 1 4 2 3 1
25
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4 2 3 4 1

• ????
• 4 2 1 3

2 4 3 1 4 2 3 1 4 2 1 3
26
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4 2 3 4 1

• ????
• 2 4 1 3

2 4 3 1 4 2 3 1 4 2 1 3 2 4 1 3
27
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4 2 3 4 1

• ????
• 2 1 4 3

2 4 3 1 4 2 3 1 4 2 1 3 2 4 1 3 2 1 4 3
28
Example
1 2 3 4 1 2 4 3 1 4 2 3 4 1 2 3 4 1 3 2 1 4 3 2 1
3 4 2 1 3 2 4 3 1 2 4 3 1 4 2 3 4 1 2 4 3 1 2 4 3
2 1 3 4 2 1 3 2 4 1 3 2 1 4 2 3 1 4 2 3 4 1

• ????
• 2 1 3 4

2 4 3 1 4 2 3 1 4 2 1 3 2 4 1 3 2 1 4 3 2 1 3 4