Question

Say that I have an array like the following:

Array
(
    [arm] => Array
        (
            [0] => A
            [1] => B
            [2] => C
        )
    [gender] => Array
        (
            [0] => Female
            [1] => Male
        )
    [location] => Array
        (
            [0] => Vancouver
            [1] => Calgary
        )
)

How can I find the cartesian product while preserving the keys of the outer associative array and using them in the inner ones? The result of the algorithm should be this:

Array
(
    [0] => Array
        (
            [arm] => A
            [gender] => Female
            [location] => Vancouver
        )

    [1] => Array
        (
            [arm] => A
            [gender] => Female
            [location] => Calgary
        )

    [2] => Array
        (
            [arm] => A
            [gender] => Male
            [location] => Vancouver
        )

...etc.

I’ve looked up quite a number of cartesian product algorithms but I’m getting stuck on the specifics of how to preserve the associative keys. The current algorithm I am using gives numerical indices only:

    $result = array();
    foreach ($map as $a) {
        if (empty($result)) {
            $result = $a;
            continue;
        }
        $res = array();
        foreach ($result as $r) {
            foreach ($a as $v) {
                $res[] = array_merge((array)$r, (array)$v);
            }
        }
        $result = $res;
    }

    print_r($result);

Any help would be appreciated.

Answer 1

Rationale

Assume that we have an input array $input with N sub-arrays, as in your example. Each
sub-array has Cn items, where n is its index inside $input, and its key is Kn. I will refer to the ith item of the nth sub-array as Vn,i.

The algorithm below can be proved to work (barring bugs) by induction:

1. For N = 1, the cartesian product is simply array(0 => array(K1 => V1,1), 1 => array(K1 => V1,2), ... ) — C1 items in total. This can be done with a simple foreach.

2. Assume that $result already holds the cartesian product of the first N-1 sub-arrays. The cartesian product of $result and the Nth sub-array can be produced this way:

3. In each item (array) inside $product, add the value KN => VN,1. Remember the resulting item (with the added value); I ‘ll refer to it as $item.

4a) For each array inside $product:

4b) For each value in the set VN,2 ... VN,CN, add to $product a copy of $item, but change the value with the key KN to VN,m (for all 2 <= m <= CN).

The two iterations 4a (over $product) and 4b (over the Nth input sub-array) ends up with $result having CN items for every item it had before the iterations, so in the end $result indeed contains the cartesian product of the first N sub arrays.

Therefore the algorithm will work for any N.

Code

function cartesian($input) {
    $result = array();

    while (list($key, $values) = each($input)) {
        // If a sub-array is empty, it doesn't affect the cartesian product
        if (empty($values)) {
            continue;
        }

        // Seeding the product array with the values from the first sub-array
        if (empty($result)) {
            foreach($values as $value) {
                $result[] = array($key => $value);
            }
        }
        else {
            // Second and subsequent input sub-arrays work like this:
            //   1. In each existing array inside $product, add an item with
            //      key == $key and value == first item in input sub-array
            //   2. Then, for each remaining item in current input sub-array,
            //      add a copy of each existing array inside $product with
            //      key == $key and value == first item of input sub-array

            // Store all items to be added to $product here; adding them
            // inside the foreach will result in an infinite loop
            $append = array();

            foreach($result as &$product) {
                // Do step 1 above. array_shift is not the most efficient, but
                // it allows us to iterate over the rest of the items with a
                // simple foreach, making the code short and easy to read.
                $product[$key] = array_shift($values);
                
                // $product is by reference (that's why the key we added above
                // will appear in the end result), so make a copy of it here
                $copy = $product;

                // Do step 2 above.
                foreach($values as $item) {
                    $copy[$key] = $item;
                    $append[] = $copy;
                }

                // Undo the side effecst of array_shift
                array_unshift($values, $product[$key]);
            }

            // Out of the foreach, we can add to $results now
            $result = array_merge($result, $append);
        }
    }

    return $result;
}

Usage

$input = array(
    'arm' => array('A', 'B', 'C'),
    'gender' => array('Female', 'Male'),
    'location' => array('Vancouver', 'Calgary'),
);

print_r(cartesian($input));