## Archive for the ‘Overlapping Sets’ Category

### Fast algorithm for successively merging k-overlapping sets?

Saturday, April 20th, 2013

Fast algorithm for successively merging k-overlapping sets?

As posted:

Consider the following algorithm for clustering sets: Begin with n sets, S1, S2,…,Sn, such that sum_{i = 1}^n |Si| = m, and successively merge sets with at least k elements in common. E.g., if S1 = {1, 2, 3}, S2 = {3, 4, 5}, and S3 = {5, 6, 7}, and k = 1 then S1 can be merged with S2 to create S1′ = {1, 2, 3, 4, 5}, and S1′ can be merged with S3 to create S1” = {1,…,7}.

Warmup question: Can this clustering algorithm be implemented efficiently for k = 1?

Answer to warmup question: If the sets only need to overlap by one element to be merged as in the example above, the clustering can be performed in O(m) time using connected components if you are careful.

Harder question: Suppose the sets must overlap by at least 2 (or k) elements to be merged. Is there an efficient algorithm for this case (i.e., close to linear time)? The challenge here is that you can have cases like S1 = {1, 2, 3}, S2 = {2, 4, 5}, S3 = {1, 4, 5}, with k = 2. Note that in this case S1 can be merged with S2 and S3, but only after S2 and S3 are merged to create S2′ so that S1 and S2′ share both elements 1 and 2.

I saw this on Theoretical Computer Science Stack Exchange earlier today.

Reminded me of overlapping set members test for [subject identifiers], [item identifiers], [subject locators], [subject identifiers] and [item identifiers], property of the other, [reified] properties. (Well, reified is a simple match, not a set.)

I have found some early work on the overlapping set member question but also work on other measures of similarity on set members.

Working up a list of papers.