1. Generating Permutations & Combinations: Selected Exercises

2. 2 10 Develop an algorithm for generating the r-permutations of a set of n elements.

3. 3 10 Solution We have algorithms to: A)Generate the next permutation in lexicographic order B)Generate the next r-combination in lexicographic order. From these, we create an algorithm to generate the r-permutations of a set with n elements: 1.Generate each r-combination, using algorithm B) 2.For each r-combination Generate the (r!) r-permutations, using algorithm A)

4. 4 10 Solution continued // pseudo code of an iterator for r-permutations. for ( Iterator ci = set.combinationIt(n,r); ci.hasNext(); ) { Set s = ci.next(); for( Iterator pi = s.permutationIt(r), pi.hasNext(); ) { int[] permutation = (int[]) pi.next(); }

5. 5 10 continue On the next slide, I put a crude Java “Iterator” for generating r-combinations based on the algorithm in the textbook. (The previous slide does not use this.)

6. 6 –// Assumption: 0 <= r <= n –public class CombinationIterator –{ – private int n; // the size of the set – private int r; // the size of the combination – private int[] combination; – private boolean hasNext = true; – private boolean isFirst = true; – – public CombinationIterator( int n, int r ) { – this.n = n; – this.r = r; – combination = new int[r]; – for ( int i = 0; i < combination.length; i++ ) – combination[i] = i + 1; – } – public boolean hasNext() { return hasNext; }

7. 7 – public int[] next() { – if ( isFirst ) { – isFirst = false; – if ( r == 0 || n <= r || n == 0 ) hasNext = false; – return combination; – } – int i = combination.length - 1; – – // find 1st submaximal element from the right – for ( ; combination[i] == n - r + i + 1; i--); – – combination[i] = combination[i] + 1; // increase that element – – // minimize subsequent elements – for ( int j = i + 1; j < combination.length; j++ ) – combination[j] = combination[i] + j - i; – – // set hasNext – for ( ; i >= 0 && combination[i] == n - r + i + 1; i--); – if ( i < 0 ) hasNext = false; – – return combination; – }

8. 8 Exercise Complete an “Iterator” class for permutations: class PermutationIterator { public PermutationIterator( int n ) boolean hasNext() int[] next() void remove() { /* null body */ } }

9. 9 Characters   . ≥ ≡ ~ ┌ ┐ └ ┘        ≈      Ω Θ     Σ ¢        