Given an array such as the following
$array = ('1', '2', '3', '4', '5', '6', '7');
I'm looking for a method to generate all possible combinations, with a minimum number of elements required in each combination r. (eg if r = 5 then it will return all possible combinations containing at least 5 elements)
A combination can be expressed as
nCr = n! / (r! - (n - r)!)
First, we determine $n as the number of elements in the array. And $r is the minimum number of elements in each combination.
$a = ['1', '2', '3', '4', '5', '6', '7']; // the array of elements we are interested in
// Determine the `n` and `r` in nCr = n! / (r! * (n-r)!)
$r = 5;
$n = count($a);
Next, we determine $max as the maximum number that can be represented by $n binary digits. That is, if $n = 3, then $max = (111)2 = 7. To do this, we first create a empty string $maxBinary and add $n number of 1s to it. We then convert it to decimal, and store it in $max.
$maxBinary = "";
for ($i = 0; $i < $n; $i++)
{
$maxBinary .= "1";
}
$max = bindec($maxBinary); // convert it into a decimal value, so that we can use it in the following for loop
Then, we list out every binary number from 0 to $max and store those that have more than $r number of 1s in them.
$allBinary = array(); // the array of binary numbers
for ($i = 0; $i <= $max; $i++)
{
if (substr_count(decbin($i), "1") >= $r) // we count the number of ones to determine if they are >= $r
{
// we make the length of the binary numbers equal to the number of elements in the array,
// so that it is easy to select elements from the array, based on which of the digits are 1.
// we do this by padding zeros to the left.
$temp = str_pad(decbin($i), $n, "0", STR_PAD_LEFT);
$allBinary[] = $temp;
}
}
Then, we use the same trick as above to select elements for our combination. I believe the comments explain enough.
$combs = array(); // the array for all the combinations.
$row = array(); // the array of binary digits in one element of the $allBinary array.
foreach ($allBinary as $key => $one)
{
$combs[$key] = "";
$row = str_split($one); // we store the digits of the binary number individually
foreach ($row as $indx => $digit)
{
if ($digit == '1') // if the digit is 1, then the corresponding element in the array is part of this combination.
{
$combs[$key] .= $a[$indx]; // add the array element at the corresponding index to the combination
}
}
}
And that is it. You are done!
Now if you have something like
echo count($combs);
then it would give you 29.
I read up on this only after seeing your question, and as a newcomer, I found these useful:
Also, here are some quick links to the docs, that should help people who see this in the future:
Combinations of k out of n items can be defined recursively using the following function:
function combinationsOf($k, $xs){
if ($k === 0)
return array(array());
if (count($xs) === 0)
return array();
$x = $xs[0];
$xs1 = array_slice($xs,1,count($xs)-1);
$res1 = combinationsOf($k-1,$xs1);
for ($i = 0; $i < count($res1); $i++) {
array_splice($res1[$i], 0, 0, $x);
}
$res2 = combinationsOf($k,$xs1);
return array_merge($res1, $res2);
}
The above is based on the recursive definition that to choose k out n elements, one can fix an element x in the list, and there are C(k-1, xs\{x}) combinations that contain x (i.e. res1), and C(k,xs\{xs}) combinations that do not contain x (i.e. res2 in code).
Full example:
$array = array('1', '2', '3', '4', '5', '6', '7');
function combinationsOf($k, $xs){
if ($k === 0)
return array(array());
if (count($xs) === 0)
return array();
$x = $xs[0];
$xs1 = array_slice($xs,1,count($xs)-1);
$res1 = combinationsOf($k-1,$xs1);
for ($i = 0; $i < count($res1); $i++) {
array_splice($res1[$i], 0, 0, $x);
}
$res2 = combinationsOf($k,$xs1);
return array_merge($res1, $res2);
}
print_r ($array);
print_r(combinationsOf(5,$array));
//print_r(combinationsOf(5,$array)+combinationsOf(6,$array)+combinationsOf(7,$array));
function arrToBit(Array $element) {
$bit = '';
foreach ($element as $e) {
$bit .= '1';
}
$length = count($element);
$num = bindec($bit);
$back = [];
while ($num) {
$back[] = str_pad(decbin($num), $length, '0', STR_PAD_LEFT);
$num--;
}
//$back[] = str_pad(decbin(0), $length, '0', STR_PAD_LEFT);
return $back;
}
function bitToArr(Array $element, $bit) {
$num = count($element);
$back = [];
for ($i = 0; $i < $num; $i++) {
if (substr($bit, $i, 1) == '1') {
$back[] = $element[$i];
}
}
return $back;
}
$tags = ['a', 'b', 'c'];
$bits = arrToBit($tags);
$combination = [];
foreach ($bits as $b) {
$combination[] = bitToArr($tags, $b);
}
var_dump($combination);
