1. Generating Permutations & Combinations: Selected Exercises

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

3. 10 Solution We have algorithms to:
Generate the next permutation in lexicographic order
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:
Generate each r-combination, using algorithm B)
For each r-combination
Generate the (r!) r-permutations, using algorithm A)

4. 10 Solution continued
// pseudo code of an iterator for r-permutations.
for ( Iterator<Set> 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. 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. // 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. 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;

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

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