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This shortest common supersequence problem is closely related to the longest common subsequence problem. Given two sequences X = < x_{1},...,x_{m} > and Y = < y_{1},...,y_{n} >, a sequence U = < u_{1},...,u_{k} > is a common supersequence of X and Y if U is a supersequence of both X and Y. ## References

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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.

- 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.

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Last updated on Saturday April 19, 2008 at 08:06:19 PDT (GMT -0700)

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