Shortest Common Superstring: find shortest string that contains all given string fragments

Question

Given some string fragments, I would like to find the shortest possible single string ("output string") that contains all the fragments. Fragments can overlap each other in the output string.

Example:

For the string fragments:

BCDA
AGF
ABC

The following output string contains all fragments, and was made by naive appending:

BCDAAGFABC

However this output string is better (shorter), as it employs overlaps:

ABCDAGF
^
ABC
 ^
 BCDA
 ^ 
 AGF

I'm looking for algorithms for this problem. It's not absolutely important to find the strictly shortest output string, but the shorter the better. I'm looking for an algorithm better than the obvious naive one that would try appending all permutations of the input fragments and removing overlaps (which would appear to be NP-Complete).

I've started work on a solution and it's proving quite interesting; I'd like to see what other people might come up with. I'll add my work-in-progress to this question in a while.

• superM: The problem seems to be NP-complete. If so, you won't be able to find a polynomial algorithm for determining the shortest string at all, but there might be polynomial algorithms which give approximate (not the shortest possible) solutions.
• occulus: This blog post regarding NP-Complete is nice: codinghorror.com/blog/2008/11/…
• superM: The blog's really nice, I read it all the time )))
• ratchet freak: @superM this is similar enough to traveling salesman (each string a city and cost between cities = some number-overlap)
• superM: @ratchet freak, it is _ you could give small cost between cities if they have more common letters, and the biggest cost when they don't have any common letter at all

Answer 1

What you're asking about is the Shortest Common Superstring problem, for which there is no algorithm that works for all cases. But it is a common problem (in compression and DNA sequencing) and several approximation algorithms are well-known.

"Greedy" algorithms are generally accepted to be the most effective (as in, they have the least-bad worst-case).

Have a read of the paper Approximation Algorithms for the Shortest Common Superstring Problem by Jonathan Turner for much more information.

• occulus: Some relevant pages: update.uu.se/~shikaree/Westling and cs.sunysb.edu/~algorith/files/shortest-common-superstring.shtml
• occulus: Hmm, note that the first link in my comment just above address supersequences and not superstrings! A supersequence does not seem to require all characters in a sequence be contiguous.
• Majid: Your link is dead.