This one reminds me of an old puzzle problem.

**The problem: **String-To-String Correction. This is problem SR20 in the appendix.

**The description: **Given two strings x, and y over a finite alphabet ∑, and an integer K. Can we start from x and derive y in K or fewer steps, where each step is either deleting a symbol or interchanging a symbol.

**Example: **The puzzles I’m thinking of are the ones that say “Can you go from HELLO to OLE in 6 steps?”

- HELLO
- ELLO
- ELO
- EOL
- OEL
- OLE

The reason this is a hard problem is when you have repeated symbols, and need to “choose” which ones to delete. A slightly harder example is to go from ABABABABBB to BAAB. Now which A’s and B’s you choose to delete have an effect on the total number of moves.

The paper by Wagner that has the reduction makes the point that if you replace “Delete” with “Insert”, you get basically the same problem but go from y to x instead of from x to y. The paper gives other variants that have polynomial solutions (allowing insertions and deletions and changes of a character, whether or not you allow interchange).

**Reduction: **Wagner uses Set Covering. So we’re given a collection of sets c_{1}..c_{n} all subsets of some set W, and an integer K.

Define t = |W| and r = t^{2}. Pick 3 symbols Q,R, and S that are not in W. The string x will be:

Q^{r}Rc_{i}Q^{r}S^{r+1}

..concatenated together each time for each c_{i}. (So the copy that has c_{1} is first, then another copy with c_{2}, and so on.

The string y will be n copies of the substring RQ^{r}, followed by w, followed by n copies of the substring S^{r+1}

The K’ for this problem is (K+1)*r-1 + 2t(r-1) + n(n-1)(r+1)^{2}/2 + d.

d is r*n+|c_{1}…c_{2}| – |W|.

From here, Wagner explains that the way the correction problem works is by deciding which copies of the R, Q^{r} and S^{r+1} sets in x match to the corresponding ones in y. By counting all of the operations that “have” to be done (by deleting or moving characters), he shows that the a correction problem that has weight of K’ has to cross over certain c_{i} sets to make a covering, and that there have to be K or less of those sets.

**Difficulty: 8**. The construction is pretty crazy. You can eventually see where the components come from, but I can’t imagine how he came up with these things (especially the x and y strings) in the first place.