Generating Permutations

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 ( IteratorltSetgt 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 lt r lt 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 intr
• for ( int i 0 i lt combination.length
  i )
• combinationi i 1
• public boolean hasNext() return hasNext

7

• public int next()
• if ( isFirst )
• isFirst false
• if ( r 0 n lt r n 0 )
  hasNext false
• return combination
• int i combination.length - 1
• // find 1st submaximal element from the
  right
• for ( combinationi n - r i 1
  i--)
• combinationi combinationi 1 //
  increase that element
• // minimize subsequent elements
• for ( int j i 1 j lt
  combination.length j )
• combinationj combinationi j -
  i
• // set hasNext
• for ( i gt 0 combinationi n - r
  i 1 i--)

8
Exercise

• Complete an Iterator class for permutations
• class PermutationIterator
• public PermutationIterator( int n )
• boolean hasNext()
• int next()
• void remove() / null body /

9
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