Getting the closest string match

Question

I need a way to compare multiple strings to a test string and return the string that closely resembles it:

TEST STRING: THE BROWN FOX JUMPED OVER THE RED COW         


        

Answer 1

This problem turns up all the time in bioinformatics. The accepted answer above (which was great by the way) is known in bioinformatics as the Needleman-Wunsch (compare two strings) and Smith-Waterman (find an approximate substring in a longer string) algorithms. They work great and have been workhorses for decades.

But what if you have a million strings to compare? That's a trillion pairwise comparisons, each of which is O(n*m)! Modern DNA sequencers easily generate a billion short DNA sequences, each about 200 DNA "letters" long. Typically, we want to find, for each such string, the best match against the human genome (3 billion letters). Clearly, the Needleman-Wunsch algorithm and its relatives will not do.

This so-called "alignment problem" is a field of active research. The most popular algorithms are currently able to find inexact matches between 1 billion short strings and the human genome in a matter of hours on reasonable hardware (say, eight cores and 32 GB RAM).

Most of these algorithms work by quickly finding short exact matches (seeds) and then extending these to the full string using a slower algorithm (for example, the Smith-Waterman). The reason this works is that we are really only interested in a few close matches, so it pays off to get rid of the 99.9...% of pairs that have nothing in common.

How does finding exact matches help finding inexact matches? Well, say we allow only a single difference between the query and the target. It is easy to see that this difference must occur in either the right or left half of the query, and so the other half must match exactly. This idea can be extended to multiple mismatches and is the basis for the ELAND algorithm commonly used with Illumina DNA sequencers.

There are many very good algorithms for doing exact string matching. Given a query string of length 200, and a target string of length 3 billion (the human genome), we want to find any place in the target where there is a substring of length k that matches a substring of the query exactly. A simple approach is to begin by indexing the target: take all k-long substrings, put them in an array and sort them. Then take each k-long substring of the query and search the sorted index. Sort and search can be done in O(log n) time.

But storage can be a problem. An index of the 3 billion letter target would need to hold 3 billion pointers and 3 billion k-long words. It would seem hard to fit this in less than several tens of gigabytes of RAM. But amazingly we can greatly compress the index, using the Burrows-Wheeler transform, and it will still be efficiently queryable. An index of the human genome can fit in less than 4 GB RAM. This idea is the basis of popular sequence aligners such as Bowtie and BWA.

Alternatively, we can use a suffix array, which stores only the pointers, yet represents a simultaneous index of all suffixes in the target string (essentially, a simultaneous index for all possible values of k; the same is true of the Burrows-Wheeler transform). A suffix array index of the human genome will take 12 GB of RAM if we use 32-bit pointers.

The links above contain a wealth of information and links to primary research papers. The ELAND link goes to a PDF with useful figures illustrating the concepts involved, and shows how to deal with insertions and deletions.

Finally, while these algorithms have basically solved the problem of (re)sequencing single human genomes (a billion short strings), DNA sequencing technology improves even faster than Moore's law, and we are fast approaching trillion-letter datasets. For example, there are currently projects underway to sequence the genomes of 10,000 vertebrate species, each a billion letters long or so. Naturally, we will want to do pairwise inexact string matching on the data...