Question

Given an array of elements in PHP, I wish to create a new two-dimensional array containing only those elements of the power set that are a specific length. As an example, for the following array:

array(4) {
    0 => 'A',
    1 => 'B',
    2 => 'C',
    3 => 'D'
}

If I were to run the function fixed_length_power_set( $arr, 2 ) then I want it to return:

array(6) {
    0 => array(2) {
        0 => 'A',
        1 => 'B'
    }
    1 => array(2) {
        0 => 'A',
        1 => 'C'
    }

    2 => array(2) {
        0 => 'A',
        1 => 'D'
    }
    3 => array(2) {
        0 => 'B',
        1 => 'C'
    }
    4 => array(2) {
        0 => 'B',
        1 => 'D'
    }
    5 => array(2) {
        0 => 'C',
        1 => 'D'
    }
}

Although I can think of a few rules to generalize the process, for some reason I can not seem to turn it into code. Does anyone have suggestions?

Answer 1

Use a simple recursive algorithm: For the set of all subsets of size k from a set of size n,

• if n == k, return a set containing the entire set;

• if k == 1 return the set of all singletons;

• otherwise remove an element x from the set: now you need all the subsets of size k-1 of the remaining set (i.e. those subsets which include x), as well as all the subsets of size k of the remaining set (those which don’t include x).

In PHP pseudo-code:

function subsets_n($arr, $k)
{
  if (count($arr) < $k) return array();
  if (count($arr) == $k) return array(0 => $arr);

  $x = array_pop($arr);
  if (is_null($x)) return array();

  return array_merge(subsets_n($arr, $k),
                     merge_into_each($x, subsets_n($arr, $k-1)) );
}

Here merge_into_each() adds x to each array in the collection:

function merge_into_each($x, $arr)
{
  foreach ($arr as &$a) array_push($a, $x);
  return $arr;
}