How would you code an anti plagiarism site?

Good plagiarism detection will apply heuristics based on the type of document (e.g. an essay or program code in a specific language).

However, you can also apply a general solution. Have a look at the Normalized Compression Distance (NCD). Obviously you cannot exactly calculate a text's Kolmogorov complexity, but you can approach it be simply compressing the text.

A smaller NCD indicates that two texts are more similar. Some compression algorithms will give better results than others. Luckily PHP provides support for several compression algorithms, so you can have your NCD-driven plagiarism detection code running in no-time. Below I'll give example code which uses Zlib:

PHP:

function ncd($x, $y) { 
  $cx = strlen(gzcompress($x));
  $cy = strlen(gzcompress($y));
  return (strlen(gzcompress($x . $y)) - min($cx, $cy)) / max($cx, $cy);
}   

print(ncd('this is a test', 'this was a test'));
print(ncd('this is a test', 'this text is completely different'));

Python:

>>> from zlib import compress as c
>>> def ncd(x, y): 
...     cx, cy = len(c(x)), len(c(y))
...     return (len(c(x + y)) - min(cx, cy)) / max(cx, cy) 
... 
>>> ncd('this is a test', 'this was a test')
0.30434782608695654
>>> ncd('this is a test', 'this text is completely different')
0.74358974358974361

Note that for larger texts (read: actual files) the results will be much more pronounced. Give it a try and report your experiences!

Well, you first of all have to understand what you're up against.

Word-for-word plagiarism should be ridiculously easy to spot. The most naive approach would be to take word tuples of sufficient length and compare them against your corpus. The sufficient length can be incredibly low. Compare Google results:

"I think" => 454,000,000
"I think this" => 329,000,000
"I think this is" => 227,000,000
"I think this is plagiarism" => 5

So even with that approach you have a very high chance to find a good match or two (fun fact: most criminals are really dumb).

If the plagiarist used synonyms, changed word ordering and so on, obviously it gets a bit more difficult. You would have to store synonyms as well and try to normalise grammatical structure a bit to keep the same approach working. The same goes for spelling, of course (i.e. try to match by normalisation or try to account for the deviations in your matching, as in the NCD approaches posted in the other answers).

However the biggest problem is conceptual plagiarism. That is really hard and there are no obvious solutions without parsing the semantics of each sentence (i.e. sufficiently complex AI).

The truth is, though, that you only need to find SOME kind of match. You don't need to find an exact match in order to find a relevant text in your corpus. The final assessment should always be made by a human anyway, so it's okay if you find an inexact match.

Plagiarists are mostly stupid and lazy, so their copies will be stupid and lazy, too. Some put an incredible amount of effort into their work, but those works are often non-obvious plagiarism in the first place, so it's hard to track down programmatically (i.e. if a human has trouble recognising plagiarism with both texts presented side-by-side, a computer most likely will, too). For all the other 80%-or-so, the dumb approach is good enough.

I think that this problem is complicated, and doesn't have one best solution. You can detect exact duplication of words at the whole document level (ie someone downloads an entire essay from the web) all the way down to the phrase level. Doing this at the document level is pretty easy - the most trivial solution would take the checksum of each document submitted and compare it against a list of checksums of known documents. After that you could try to detect plagiarism of ideas, or find sentences that were copied directly then changed slightly in order to throw off software like this.

To get something that works at the phrase level you might need to get more sophisticated if want any level of efficiency. For example, you could look for differences in style of writing between paragraphs, and focus your attention to paragraphs that feel "out of place" compared to the rest of a paper.

There are lots of papers on this subject out there, so I suspect there is no one perfect solution yet. For example, these 2 papers give introductions to some of the general issues with this kind of software,and have plenty of references that you could dig deeper into if you'd like.

http://ir.shef.ac.uk/cloughie/papers/pas_plagiarism.pdf

http://proceedings.informingscience.org/InSITE2007/IISITv4p601-614Dreh383.pdf