Generating Permutations & Combinations: Selected Exercises

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Develop an algorithm for generating the

r-permutations of a set of n elements.

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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)

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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();

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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.)

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– // 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;

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– 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;– –

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Exercise

Complete an “Iterator” class for permutations:class PermutationIterator

public PermutationIterator( int n )

boolean hasNext()

int[] next()

void remove() /* null body */

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