## Introduction

In my previous article, 'A C# Permutation Iterator', I presented a compact way to permutate over a sequence. In this article, I will expand my iterator to include combinations.

## Using the Code

This iterator works recursively. The recursion terminal condition is when we elect to choose 0 elements, in which case the sequence is returned as is. In the recursive case, it is a bit more complex than the permutation iterator. The algorithm works by holding the first element, and making a recursive call on the rest of the sequence. This is done iteratively (n) times where (n) is the length of the sequence. However, each progressive iteration calls the recursion with an ever decreasing sequence length. As an example, here is a depiction of the process when choosing 2 elements from a sequence of 5 (the result of each iteration is the first 2 elements):

Sequence yielded | Comments | Result (the first 2 elements) |

12345 | The initial sequence and first result | 12 |

13452 | Inner rotation (starting at the 2nd element) | 13 |

14523 | Inner rotation | 14 |

15234 | Inner rotation | 15 |

23451 | Outer rotation | 23 |

24531 | Inner rotation - Note that the last element is now excluded from the rotation | 24 |

25341 | Inner rotation | 25 |

34512 | Outer rotation | 34 |

35412 | Inner rotation - Note how the last two elements are now excluded from the rotation | 35 |

45123 | Outer rotation | 45 |

And here is the code:

private static void RotateLeft<T>(IList<T> sequence, int start, int count)
{
T tmp = sequence[start];
sequence.RemoveAt(start);
sequence.Insert(start + count - 1, tmp);
}
public static IEnumerable<IList<T>> Combinations<T>(
IList<T> sequence,
int start,
int count,
int choose)
{
if (choose == 0) yield return sequence;
else
{
for (int i = 0; i < count; i++)
{
foreach (var perm in Combinations(
sequence,
start + 1,
count - 1 - i,
choose - 1))
yield return perm;
RotateLeft(sequence, start, count);
}
}
}

Note that each iteration returns a full length sequence, however the result is actually contained in the first (m) elements, where (m) is the number of elements we elected to choose from the sequence.

The initial call to the `Combinations `

function should pass `0 `

as the `start`

parameter and the sequence length as the `count`

parameter. To make it easier, we can overload the call thus:

public static IEnumerable<IList<T>> Combinations<T>(
IList<T> sequence,
int choose)
{
return Combinations(sequence, 0, sequence.Count, choose);
}

Following is an example of iterating over combinations of 3 characters over the `string `

"`abcdef`

". Note how I am using the `Take`

extension method to grab only the result part from the iteration result.

private static void CombinationsOverString(string s, int count)
{
foreach (var comb in Iterator.Combinations(s.ToCharArray().ToList(), count))
{
string r = new string(comb.Take(count).ToArray());
Console.Write("{0,-8}", r);
}
Console.WriteLine();
}

Output:

abc abd abe abf acd ace acf ade adf aef
bcd bce bcf bde bdf bef cde cdf cef def

And here is an example of permutations over these combinations using the permutation iterator from my previous article.

private static void CombinationsPermutations(string s, int count)
{
foreach (var combo in Iterator.Combinations(
s.ToCharArray().ToList(),
count))
{
foreach (var permu in Iterator.Permutations(combo, count))
{
string r = new string(permu.Take(count).ToArray());
Console.Write("{0,-8}", r);
}
}
Console.WriteLine();
}

Output:

abc bac cab acb bca cba abd bad dab adb
bda dba abe bae eab aeb bea eba abf baf
fab afb bfa fba acd cad dac adc cda dca
ace cae eac aec cea eca acf caf fac afc
cfa fca ade dae ead aed dea eda adf daf
fad afd dfa fda aef eaf fae afe efa fea
bcd cbd dbc bdc cdb dcb bce cbe ebc bec
ceb ecb bcf cbf fbc bfc cfb fcb bde dbe
ebd bed deb edb bdf dbf fbd bfd dfb fdb
bef ebf fbe bfe efb feb cde dce ecd ced
dec edc cdf dcf fcd cfd dfc fdc cef ecf
fce cfe efc fec def edf fde dfe efd fed

## Points of Interest

There are many implementations of combination and permutation iterators to be found on the internet, however I particularly like this one due to the fact that it is light on memory, and doesn't create any unnecessary objects. The corollary of this, is that the sequence is modified while the iterator is running; if this is not desired, the iteration should be run on a copy of the original sequence.

## History

- 16-Nov-2009 - Initial posting