Similar String algorithm

Question

I'm looking for an algorithm, or at least theory of operation on how you would find similar text in two or more different strings...

Much like the question posed here: http://stackoverflow.com/questions/246961/algorithm-to-find-similar-text, the difference being that my text strings will only ever be a handful of words.

Like say I have a string: "Into the clear blue sky" and I'm doing a compare with the following two strings: "The color is sky blue" and "In the blue clear sky"

I'm looking for an algorithm that can be used to match the text in the two, and decide on how close they match. In my case, spelling, and punctuation are going to be important. I don't want them to affect the ability to discover the real text. In the above example, if the color reference is stored as "'sky-blue'", I want it to still be able to match. However, the 3rd string listed should be a BETTER match over the second, etc.

I'm sure places like Google probably use something similar with the "Did you mean:" feature...

* EDIT *
In talking with a friend, he worked with a guy who wrote a paper on this topic. I thought I might share with with everyone reading this, as there are some really good methods and processes described in it...

Here's the link to his paper, I hope it can offer someone some help on this topic.

Answer 1

One way (although this is perhaps better suited a spellcheck-type algorithm) is the "edit distance", ie., calculate how many edits it takes to transform one string to another. A common technique is found here:

http://en.wikipedia.org/wiki/Levenshtein_distance

Answer 2

Levenshtein distance will not completely work, because you want to allow rearrangements. I think your best bet is going to be to find best rearrangement with levenstein distance as cost for each word.

To find the cost of rearrangement, kinda like the pancake sorting problem. So, you can permute every combination of words (filtering out exact matches), with every combination of other string, trying to minimize a combination of permute distance and Levenshtein distance on each word pair.

edit: Now that I have a second I can post a quick example (all 'best' guesses are on inspection and not actually running the algorithms):

original strings             | best rearrangement w/ lev distance per word
Into the clear blue sky      |    Into the c_lear blue sky 
The color is sky blue        |    is__ the colo_r blue sky

R_dist = dist( 3 1 2 5 4 ) --> 3 1 2 *4 5* --> *2 1 3* 4 5 --> *1 2* 3 4 5 = 3  
L_dist = (2D+S) + (I+D+S) (Total Subsitutions: 2, deletions: 3, insertion: 1)

(notice all the flips include all elements in the range, and I use ranges where Xi - Xj = +/- 1)

Other example

original strings             | best rearrangement w/ lev distance per word
Into the clear blue sky      |   Into the clear blue sky 
In the blue clear sky        |   In__ the clear blue sky

R_dist = dist( 1 2 4 3 5 ) -->  1 2 *3 4* 5  = 1
L_dist = (2D) (Total Subsitutions: 0, deletions: 2, insertion: 0)

And to show all possible combinations of the three...

The color is sky blue         |    The colo_r is sky blue
In the blue clear sky         |    the c_lear in sky blue

R_dist = dist( 2 4 1 3 5 ) --> *2 3 1 4* 5 --> *1 3 2* 4 5 --> 1 *2 3* 4 5 = 3
L_dist = (D+I+S) + (S) (Total Subsitutions: 2, deletions: 1, insertion: 1)

Anyway you make the cost function the second choice will be lowest cost, which is what you expected!

Answer 3

Possible approach:

Construct a Dictionary with a string key of "word1|word2" for all combinations of words in the reference string. A single combination may happen multiple times, so the value of the Dictionary should be a list of numbers, each representing the distance between the words in the reference string.

When you do this, there will be duplication here: for every "word1|word2" dictionary entry, there will be a "word2|word1" entry with the same list of distance values, but negated.

For each combination of words in the comparison string (words 1 and 2, words 1 and 3, words 2 and 3, etc.), check the two keys (word1|word2 and word2|word1) in the reference string and find the closest value to the distance in the current string. Add the absolute value of the difference between the current distance and the closest distance to a counter.

If the closest reference distance between the words is in the opposite direction (word2|word1) as the comparison string, you may want to weight it smaller than if the closest value was in the same direction in both strings.

When you are finished, divide the sum by the square of the number of words in the comparison string.

This should provide some decimal value representing how closely each word/phrase matches some word/phrase in the original string.

Of course, if the original string is longer, it won't account for that, so it may be necessary to compute this both directions (using one as the reference, then the other) and average them.

I have absolutely no code for this, and I probably just re-invented a very crude wheel. YMMV.

Answer 4

You might want to look into the algorithms used by biologists to compare DNA sequences, since they have to cope with many of the same things (chunks may be missing, or have been inserted, or just moved to a different position in the string.

The Smith-Waterman algorithm would be one example that'd probably work fairly well, although it might be too slow for your uses. Might give you a starting point, though.

Answer 5

The difficulty would be to match the strings semantically.

You could generate some kind of value based on the lexical properties of the string. e.g. They bot have blue, and sky, and they're in the same sentence, etc etc... But it won't handle cases where "Sky's jean is blue", or some other odd ball English construction that uses same words, but you'd need to parse the English grammar...

To do anything beyond lexical similarity, you'd need to look at natural language processing, and there isn't going to be one single algorith that would solve your problem.

Answer 6

One way to determine a measure of "overall similarity without respect to order" is to use some kind of compression-based distance. Basically, the way most compression algorithms (e.g. gzip) work is to scan along a string looking for string segments that have appeared earlier -- any time such a segment is found, it is replaced with an (offset, length) pair identifying the earlier segment to use. You can use measures of how well two strings compress to detect similarities between them.

Suppose you have a function string comp(string s) that returns a compressed version of s. You can then use the following expression as a "similarity score" between two strings s and t:

len(comp(s)) + len(comp(t)) - len(comp(s . t))

where . is taken to be concatenation. The idea is that you are measuring how much further you can compress t by looking at s first. If s == t, then len(comp(s . t)) will be barely any larger than len(comp(s)) and you'll get a high score, while if they are completely different, len(comp(s . t)) will be very near len(comp(s) + comp(t)) and you'll get a score near zero. Intermediate levels of similarity produce intermediate scores.

Actually the following formula is even better as it is symmetric (i.e. the score doesn't change depending on which string is s and which is t):

2 * (len(comp(s)) + len(comp(t))) - len(comp(s . t)) - len(comp(t . s))

This technique has its roots in information theory.

Advantages: good compression algorithms are already available, so you don't need to do much coding, and they run in linear time (or nearly so) so they're fast. By contrast, solutions involving all permutations of words grow super-exponentially in the number of words (although admittedly that may not be a problem in your case as you say you know there will only be a handful of words).

Answer 7

I can't mark two answers here, so I'm going to answer and mark my own. The Levenshtein distance appears to be the correct method in most cases for this. But, it is worth mentioning j_random_hackers answer as well. I have used an implementation of LZMA to test his theory, and it proves to be a sound solution. In my original question I was looking for a method for short strings (2 to 200 chars), where the Levenshtein Distance algorithm will work. But, not mentioned in the question was the need to compare two (larger) strings (in this case, text files of moderate size) and to perform a quick check to see how similar the two are. I believe that this compression technique will work well but I have to yet studied it to find at which point one becomes better than the other, in terms of the size of the sample data and the speed/cost of operation.