How to sort an array of Roman numerals?

Question

I have an array containing Roman numerals (as strings of course). Like this:

 $a = array('XIX', 'LII', 'V', 'MCCXCIV', 'III', 'XIII');

I'd like to sort them according to the numeric values of these numerals, so the results should be something like:

 $sorted_a = array('III', 'V', 'XIII', 'XIX', 'LII', 'MCCXCIV');

So my question is: what is the best way to sort an array of Roman numerals? I know how to use the array sorting functions of PHP, I'm interested in the logic that goes on inside the comparison function.

EDIT: For simplicity, I'm only looking for a way that deals with strings constructed of the basic numerals in a standard way (no CCCC for example):

I, V, X, L, C, D, M

---

TEST RESULTS

I took the time to extensively test all the code examples that were posted. Two tests were taken, one with a random array of 20 Roman numerals, and a second with an array containing 4000 of those. Same machine, lot of iterations, an average time taken, and all this run several times. Of course this is nothing offical, just my own tests.

TEST WITH 20 NUMERALS:

1. hakre, bazmegakapa - around 0.0005 s
2. anemgyenge, Andrea, Dirk McQuickly - around 0.0010 s
3. Joe Nelson - around 0.0050 s
4. Rob Hruska - around 0.0100 s

TEST WITH 4000 NUMERALS:

1. hakre, bazmegakapa - around 0.13 s
2. anemgyenge - around 1.4 s
3. Dirk McQuickly, Andrea - around 1.8 s
4. Rob Hruska - around 2.8 s
5. Joe Nelson - around 15 s (surprise, checked several more times)

I have a hard time awarding the bounty. hakre and I made the fastest versions, following the same route, but he made a variation of mine, which was previously based on borrible's idea. So I will accept hakre's solution, because that is the quickest and nicer than mine (IMO). But I will award the bounty to anemgyenge, because I love his version and a lot of effort seems to be put into it.

Answer 1

Picking your class to convert roman numbers to integers, a user-defined sort callback can handle this to sort the array:

$a = array('XIX', 'LII', 'V', 'MCCXCIV', 'III', 'XIII');

$bool = usort($a, function($a, $b) {
    return RomanNumber::Roman2Int($a) - RomanNumber::Roman2Int($b);
});    
var_dump($a);

So here you find the logic inside the comparison function: if both values are of the same weight, return 0. If the first is lower than the second, return < 0 (e.g. -1), otherwise the second is larger than the first so return > 0 (e.g. 1).

Naturally any other type of function that returns the decimal value for a roman number would work as well.

Edit:

As you commented, you do not want to run the conversion for each pair. That's fine, with a help of an additional array which contains all converted values, you can run the sort on the decimal values and use that sorting on the roman numbers as well (Demo):

$a = array('XIX', 'LII', 'V', 'MCCXCIV', 'III', 'XIII');
$b = array_map('RomanNumber::Roman2Int', $a);
array_multisort($b, $a);
var_dump($a);

array_multisort ^{PHP Manual} does most of the magic here.