Shortest common supersequence problem

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

• 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.

External links

• Dictionary of Algorithms and Data Structures: shortest common supersequence