Shortest common supersequence

This shortest common supersequence problem is closely related to the longest common subsequence problem. Given two sequences X = < x₁,...,x_m > and Y = < y₁,...,y_n >, a sequence U = < u₁,...,u_k > is a common supersequence of X and Y if U is a supersequence of both X and Y.

The shortest common supersequence (scs) is a common supersequence of minimal length. In the shortest common supersequence problem, the two sequences X and Y are given and the task is to find a shortest possible common supersequence of these sequences. In general, the scs is not unique.

For two input sequences, an scs can be formed from an lcs easily. For example, if X$[1..m]\; =\; abcbdab$ and Y$[1..n]\; =\; bdcaba$, the lcs is Z$[1..r]\; =\; bcba$. By inserting the non-lcs symbols while preserving the symbol order, we get the scs: U$[1..t]\; =\; abdcabdab$.

It is quite clear that $r\; +\; t\; =\; m\; +\; n$ for two input sequences. However, for three or more input sequences this does not hold. Note also, that the lcs and the scs problems are not dual problems.

References

* A4.2: SR8, pg.228.

External links

* [http://nist.gov/dads/HTML/shortestCommonSuperseq.html Dictionary of Algorithms and Data Structures: shortest common supersequence]