Shortest common supersequence
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This shortest common supersequence problem is closely related to the lcs (longest common subsequence problem). Again, assume that two sequences X = < x1,...,xm > and Y = < y1,...,yn > are given.
A sequence U = < u1,...,uk > 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, a scs can be formed from a 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.
[edit] References
- Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A4.2: SR8, pg.228.
