Finding the most frequent subtrees in a collection of (parse) trees

Question

I have a collection of trees whose nodes are labelled (but not uniquely). Specifically the trees are from a collection of parsed sentences (see http://en.wikipedia.org/wiki/Treebank). I wish to extract the most common subtrees from the collection - performance is not (yet) an issue. I'd be grateful for algorithms (ideally Java) or pointers to tools which do this for treebanks. Note that order of child nodes is important.

EDIT @mjv. We are working in a limited domain (chemistry) which has a stylised language so the varirty of the trees is not huge - probably similar to children's readers. Simple tree for "the cat sat on the mat".

<sentence>
  <nounPhrase>
    <article/>
    <noun/>
  </nounPhrase>
  <verbPhrase>
    <verb/>
    <prepositionPhrase>
      <preposition/>
      <nounPhrase>
        <article/>
        <noun/>
      </nounPhrase>
    </prepositionPhrase>
  </verbPhrase>
</sentence>

Here the sentence contains two identical part-of-speech subtrees (the actual tokens "cat". "mat" are not important in matching). So the algorithm would need to detect this. Note that not all nounPhrases are identical - "the big black cat" could be:

      <nounPhrase>
        <article/>
        <adjective/>
        <adjective/>
        <noun/>
      </nounPhrase>

The length of sentences will be longer - between 15 to 30 nodes. I would expect to get useful results from 1000 trees. If this does not take more than a day or so that's acceptable.

Obviously the shorter the tree the more frequent, so nounPhrase will be very common.

EDIT If this is to be solved by flattening the tree then I think it would be related to Longest Common Substring, not Longest Common Sequence. But note that I don't necessarily just want the longest - I want a list of all those long enough to be "interesting" (criterion yet to be decided).

Answer 1

I think, although you say that performance isn't yet an issue, this is an NP-hard problem, so it may never be possible to make it fast. If I've understood correctly, you can consider this a variant of the Longest common subsequence problem; if you flatten your tree into a straight sequence like

(nounphrase)(DOWN)(article:the)(adjective:big)(adjective:black)(noun:cat)(UP)

Then your problem becomes LCS.

Wikibooks has a java implementation of LCS here

Answer 2

Finding the most frequent subtrees in the collection, create a compact form of the subtree, then iterate every subtree and use a hashset to count their occurrences. 30 nodes is too big for a perfect hash - it's only about one bit per node, and you need that much to indicate whether it's a sibling or a child.

That problem isn't LCS - the most common sequence isn't related to the longest common subsequence. The most frequent subtree is that which occurs the most.

It should be at worst case O(N L^2) for N trees of length L (assuming testing equality of a subtree containing L nodes is O(L)).

Answer 3

This is a well-known problem in computer science, for which there are efficient solutions.

Here are some relevant references:

Kenji Abe, Shinji Kawasoe, Tatsuya Asai, Hiroki Arimura, Setsuo Arikawa, Optimized Substructure Discovery for Semi-structured Data, Proc. 6th European Conference on Principles and Practice of Knowledge Discovery in Databases (PKDD-2002), LNAI 2431, Springer-Verlag, 1-14, August 2002.

Mohammed J. Zaki, Efficiently Mining Frequent Trees in a Forest, 8th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, July 2002.

Or, if you just want fast code, go here: FREQT (transforming xml to S-expressions shouldn't give you too much problems, and is left as an exercise for the reader)