String-to-string correction problem

In computer science, the string-to-string correction problem refers to determining the minimum cost sequence of edit operations necessary to change one string into another (i.e., computing the shortest edit distance). Each type of edit operation has its own cost value.^[1] A single edit operation may be changing a single symbol of the string into another (cost W_C), deleting a symbol (cost W_D), or inserting a new symbol (cost W_I).^[2]

If all edit operations have the same unit costs (W_C = W_D = W_I = 1) the problem is the same as computing the Levenshtein distance of two strings.

Several algorithms exist to provide an efficient way to determine string distance and specify the minimum number of transformation operations required.^[3][4] Such algorithms are particularly useful for delta creation operations where something is stored as a set of differences relative to a base version. This allows several versions of a single object to be stored much more efficiently than storing them separately. This holds true even for single versions of several objects if they do not differ greatly, or anything in between. Notably, such difference algorithms are used in molecular biology to provide some measure of kinship between different kinds of organisms based on the similarities of their macromolecules (such as proteins or DNA).

Extension

The extended variant of the problem includes a new type of edit operation: swapping any two adjacent symbols, with a cost of W_S.

This version can be solved in polynomial time under certain restrictions on edit operation costs.^[2][5]

Robert A. Wagner (1975) showed that the general problem is NP-complete. In particular, he proved that when W_I < W_C = W_D = ∞ and 0 < W_S < ∞ (or equivalently, changing and deletion are not permitted), the problem is NP-complete.^[5]

References

1. Wagner, Robert A.; Fischer, Michael J. (1974). "The String-to-String Correction Problem". Journal of the ACM. 21 (1): 168–173. doi:10.1145/321796.321811. S2CID 13381535.
2. Lowrance, Roy; Wagner, Robert A. (April 1975). "An Extension of the String-to-String Correction Problem". Journal of the ACM. 22 (2): 177–183. doi:10.1145/321879.321880. S2CID 18892193.
3. Edit distance#Computation
4. Tichy, Walter F. (1984). "The string-to-string correction problem with block moves". ACM Transactions on Computer Systems. 2 (4): 309–321. doi:10.1145/357401.357404. S2CID 14034845.
5. Wagner, Robert A. (May 1975). "On the complexity of the Extended String-to-String Correction Problem". Proceedings of seventh annual ACM symposium on Theory of computing - STOC '75. pp. 218–223. doi:10.1145/800116.803771. ISBN 9781450374194. S2CID 18705107.