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    <p>Here's a solution I wouldn't be ashamed to show.</p> <h2>Rationale</h2> <p>Assume that we have an input array <code>$input</code> with <code>N</code> sub-arrays, as in your example. Each sub-array has <code>Cn</code> items, where <code>n</code> is its index inside <code>$input</code>, and its key is <code>Kn</code>. I will refer to the <code>i</code>th item of the <code>n</code>th sub-array as <code>Vn,i</code>.</p> <p>The algorithm below can be proved to work (barring bugs) by induction:</p> <p>1) For N = 1, the cartesian product is simply <code>array(0 =&gt; array(K1 =&gt; V1,1), 1 =&gt; array(K1 =&gt; V1,2), ... )</code> -- C1 items in total. This can be done with a simple <code>foreach</code>.</p> <p>2) Assume that <code>$result</code> already holds the cartesian product of the first N-1 sub-arrays. The cartesian product of <code>$result</code> and the Nth sub-array can be produced this way:</p> <p>3) In each item (array) inside <code>$product</code>, add the value <code>KN =&gt; VN,1</code>. Remember the resulting item (with the added value); I 'll refer to it as <code>$item</code>.</p> <p>4a) For each array inside <code>$product</code>:</p> <p>4b) For each value in the set <code>VN,2 ... VN,CN</code>, add to <code>$product</code> a copy of <code>$item</code>, but change the value with the key <code>KN</code> to <code>VN,m</code> (for all <code>2 &lt;= m &lt;= CN</code>).</p> <p>The two iterations 4a (over <code>$product</code>) and 4b (over the Nth input sub-array) ends up with <code>$result</code> having <code>CN</code> items for every item it had before the iterations, so in the end <code>$result</code> indeed contains the cartesian product of the first N sub arrays.</p> <p>Therefore the algorithm will work for any N.</p> <p><em>This was harder to write than it should have been. My formal proofs are definitely getting rusty...</em></p> <h2>Code</h2> <pre><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 =&gt; $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 &amp;$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 $result; } </code></pre> <h2>Usage</h2> <pre><code>$input = array( 'arm' =&gt; array('A', 'B', 'C'), 'gender' =&gt; array('Female', 'Male'), 'location' =&gt; array('Vancouver', 'Calgary'), ); print_r(cartesian($input)); </code></pre> <h2><a href="http://ideone.com/7mRCL" rel="noreferrer">See it in action!</a></h2>
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