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  1. PO
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    <p><em>Equivalence relations</em> are particularly nice kinds of matching; they satisfy three properties:</p> <ul> <li>reflexivity: for any value A, A ~ A</li> <li>symmetry: if A ~ B, then necessarily B ~ A</li> <li>transitivity: if A ~ B and B ~ C, then necessarily A ~ C</li> </ul> <p>What makes these nice is that they allow you to partition your data into disjoint sets such that each pair of elements in any given set are related by ~. So, what you can do is apply the union-find algorithm to first partition all your data, then pick out a single <em>representative</em> element from each set in the partition; this completely de-duplicates the data (where "duplicate" means "related by ~"). Moreover, this solution is <em>canonical</em> in the sense that no matter which representatives you happen to pick from each partition, you get the same number of final values, and each of the final values are pairwise non-duplicate.</p> <p>Unfortunately, fuzzy matching is not an equivalence relation, since it is presumably not transitive (though it's probably reflexive and symmetric). The result of this is that there isn't a canonical way to partition the data; you might find that any way you try to partition the data, some values in one set are equivalent to values from another set, or that some values from within a single set are not equivalent.</p> <p>So, what behavior do you want, exactly, in these situations?</p>
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    1. POFuzzy matching deduplication in less than exponential time?
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    1. USDaniel Wagner
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    1. COI'm not sure I understood your answer, so sorry if this isn't what you are looking for! So what you're saying is: there is no guarantee when I divide up my data based on the results of some relationship between the data, because fuzzy matching fails transitivity, there is no guarantee that the pairs created will encompass all pairs that are in fact similar (satisfy that relationship); or, all pairs that satisfy will be present, but some false positives as well. I prefer accuracy in this case, so if there is a choice, I'd go for the latter. But I probably misunderstood your answer :)
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      1. USDave W.
    2. COHere's the simplest example illustrating the dilemma. Suppose ~ is "are edit-distance 1 apart". We have the words "goo", "foo", and "food". How should they be de-duplicated? There are no duplicates in the result {"goo", "food"} (and the remainder, "foo", is a duplicate of at least one of these); there are no duplicates in the result {"foo"} (and the remainders, "goo" and "good", are each duplicates of at least one of these); but these two results are significantly different. Which one do you prefer? Can you say carefully why you prefer that one?
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      1. USDaniel Wagner
    3. COGiven the words "goo", "foo", and "food", and a "duplicate" threshold of 1 edit distance, it seems to me that "goo" and "foo" should match (aka be marked as possible duplicates) and "foo" and "food" should as well, but "goo" and "food" should not be marked as duplicates, because they are 2 edit-distance apart. But this seems relatively straightforward, so I think I might not be understanding your comment (once again!). Thanks for your patience explaining this, I really appreciate it!
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      1. USDave W.
 

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