Algorithms for Generating Permutations and Combinations

Algorithms for Generating Permutations andCombinationsSection 6.3Prof. Nathan WodarzMath 209 - Fall 2008Contents1 Listing Permutations and Combinations 21.1 Listing Permutations and Combinations . . . . . . . . . . . . . . 21.2 Lexicographic Order . . . . . . . . . . . . . . . . . . . . . . . . 22 Generating Permutations and Combinations 42.1 Generating Combinations . . . . . . . . . . . . . . . . . . . . . . 42.2 Generating Permutations . . . . . . . . . . . . . . . . . . . . . . 61

1 Listing Permutations and Combinations1.1 Listing Permutations and CombinationsListing Permutations and Combinations• Goal: List all permutations and/or combinations of a set• Problems:– Lots of them– How can we be sure all are listed?– Idea: Put some sort of order on permutations/combinations1.2 Lexicographic OrderLexicographic Order• Will use lexicographic order to list all permutations and/or combinations• Similar to dictionary (alphabetical) order– If Word A is shorter than Word B, and every letter of Word A occurs inthe same place in Word B, Word A comes before Word B (”compute”and ”computer”)– If the first letter that differs in Word A comes before the correspondingletter in Word B, then Word A comes before Word B (”math” and”matter”)• For strings α = s 1 s 2 s 3 · · · s p and β = t 1 t 2 t 3 · · · t q taken from the set {1, 2, 3, . . . , n}– For example, α = 1742 and β = 18285 are strings over {1, 2, 3, 4, 5, 6, 7, 8}– We write α < β (α is lexicographically less than β) provided that∗ p < q and s i = t i for 1 ≤ i ≤ q (e.g., α = 1732 and β = 173245)∗ For the first i such that s i t i , s i < t i (e.g., α = 28473 andβ = 2848)2

Lexicographic Order and PermutationsExample. For the following 4−permutations from the set {1, 2, 3, 4, 5, 6, 7}, findthe permutation that immediately follows them in lexicographic order1. 1234 is followed by2. 4567 is followed by3. 5437 is followed by4. 7654 is followed byLexicographic Order and Combinations• We will always list a given combinations the order s 1 < s 2 < · · · < s pExample. For the following 4−combinations from the set {1, 2, 3, 4, 5, 6, 7}, findthe combination that immediately follows them in lexicographic order1. 1234 is followed by2. 3467 is followed by3. 4567 is followed by3

2 Generating Permutations and Combinations2.1 Generating CombinationsGenerating Combinations• Given a string α = s 1 · · · s r , to find the next string (as a combination)– Find the rightmost element not at its maximum value– Don’t change anything before that element– Increment the element found above– Each additional element is one more than the previous• For 5-combinations of {1, 2, 3, 4, 5, 6, 7, 8}:– We will find successor of 13578– What is rightmost element not at its maximum?– Increase that by 1– List remaining elements in order.– Successor is4

Algorithm for Generating CombinationsList all r-combinations of {1, 2, . . . , n} in increasing lexicographic order.Input: r, nOutput: All r-combinations of {1,2,...,n}in increasing lexicographic order1. combination(r,n){2. for i = 1 to r3. s i = i// Print the first r-combination4. print(s 1 , s 2 , ..., s r )5. for i = 2 to C(n,r) {6. m = r7. max val = n8. while (s m == max val){// Find the rightmost elementnot at maximum value9. m = m - 110. max val--11. }// Increment the above rightmostelement12. s r ++// All others are the successorsof this element13. for j = m + 1 to r14. s j = s j−1 + 1// Print this new combination15. print(s 1 , s 2 , ..., s r )16. }17. }5

2.2 Generating PermutationsGenerating Permutations• Given a string α = s 1 · · · s r , to find the next string (as a permutation)– Find the rightmost place where digits increase– Don’t change anything before that element– Make the left element of the pair as small as possible but still largerthan it was– Each additional element is as small as possible• For permutations of {1, 2, 3, 4, 5, 6}:– We will find successor of 135642– What is rightmost place the digits increase?– Increase the leftmost to be smallest possible– List remaining elements in smallest to largest.– Successor is6

Algorithm for Generating PermutationsList all permutations of {1, 2, . . . , n} in increasing lexicographic order.Input: nOutput: All permutations of {1,2,...,n} in increasing lexicographic order1. permutation(n){2. for i = 1 to r3. s i = i// Print the first permutation4. print(s 1 , s 2 , ..., s r )5. for i = 2 to n! {6. m = n - 17. while (s m > s m+1 )// Find the last decrease8. m = m - 19. k = n10. while (s m > s k )// Find the last elementgreater than s m11. k = k - 112. swap(s m , s k )13. p = m + 114. q = n15. while (p < q) {// swap s m+1 and s n , swap s m+2and s n −1 , ...16. swap(s p , s q )17. p++18. q--19. }// Print this new permutation20. print(s 1 , s 2 , ..., s r )21. }22. }SummarySummaryYou should be able to:• Work with lexicographic ordering• Find the next combination and/or permutation of a given one7