Shortest common supersequence

In computer science, the shortest common supersequence problem is a problem 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. In other words, the shortest common supersequence between strings x and y is the shortest string z such that both x and y are subsequences of z.

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 a longest common subsequence (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

• Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. p. 228 A4.2: SR8. ISBN 0-7167-1045-5. Zbl 0411.68039.
• Szpankowski, Wojciech (2001). Average case analysis of algorithms on sequences. Wiley-Interscience Series in Discrete Mathematics and Optimization. With a foreword by Philippe Flajolet. Chichester: Wiley. ISBN 0-471-24063-X. Zbl 0968.68205.

External links

• Dictionary of Algorithms and Data Structures: shortest common supersequence