algorithms, not-hadoop

Super Fast Estimates of Levenshtein Distance

Levenshtein Distance is an elegant measure of the dissimilarity of two strings. Given a pair of strings, say, “hat” and “cat”, the LD is the number of single-character edits that are required to turn one into the other.  The LD of cat and hat is one. The LD of hats and cat is two.

LD is precise and has an intuitive meaning but in practice it is used mostly for short strings because the run-time of the algorithm to compute LD is quadratic, i.e, proportional to the product of  the lengths of the two strings. On a reasonable computer you can compare strings as long as a line in this article in a few microseconds, but comparing a full page of this article to another full page would take a good chunk of a second.  Comparing two books with LD is serious computing–many of minutes if you have a computer with sufficient memory, which you almost certainly do not.

That’s why, as elegant as LD is for describing how different two chunks of text are, people rarely consider using it as a way to compare documents such as Web pages, articles, or books.

This heuristic described here is a way around that problem.  It turns out that you can compute a decent estimate of the LD of two large strings many thousands of times faster than you could compute the true LD.  The other way to say this is that whatever amount of time you deem tolerable for a comparison computation, using estimates increases the size of strings you can compare within that time limit by a factor of as much as a few hundred. The practical size-range for estimated LD is in the megabyte range (and much larger for binary data.)

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Equally importantly, the LD estimates are made from pre-computed signatures, not from the the original documents. This means that it is not necessary to have the documents to be compared on hand at the time the estimate is computed, which is a tremendous advantage when you need to compare documents across a network.

The signatures can also provide insight into approximately where and how two sequences differ. This allows finer distinctions to be made about near-duplication, for instance, is one document embedded in the other, or are many small difference sprinkled throughout?

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