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| In [[computer science]], the '''shortest common supersequence problem''' is a problem closely related to the [[longest common subsequence problem]]. Given two sequences '''X''' = < x<sub>1</sub>,...,x<sub>m</sub> > and '''Y''' = < y<sub>1</sub>,...,y<sub>n</sub> >, a sequence '''U''' = < u<sub>1</sub>,...,u<sub>k</sub> > 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 [[subsequence]]s of z.
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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.
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| For two input sequences, an scs can be formed from a longest common subsequence (lcs) easily. For example, if '''X'''<math>[1..m] = abcbdab</math> and '''Y'''<math>[1..n] = bdcaba</math>, the lcs is '''Z'''<math>[1..r] = bcba</math>. By inserting the non-lcs symbols while preserving the symbol order, we get the scs: '''U'''<math>[1..t] = abdcabdab</math>.
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| It is quite clear that <math> r + t = m + n </math> 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 problem]]s.
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| == References ==
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| * {{cite book | first1=Michael R. | last1=Garey | author1-link=Michael R. Garey | first2=David S. | last2=Johnson | author2-link=David S. Johnson | year = 1979 | title = [[Computers and Intractability: A Guide to the Theory of NP-Completeness]] | publisher = W.H. Freeman | isbn = 0-7167-1045-5 | zbl=0411.68039 | at=p. 228 A4.2: SR8 }}
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| * {{cite book | last=Szpankowski | first=Wojciech | title=Average case analysis of algorithms on sequences | others=With a foreword by Philippe Flajolet | series=Wiley-Interscience Series in Discrete Mathematics and Optimization | location=Chichester | publisher=Wiley | year=2001 | isbn=0-471-24063-X | zbl=0968.68205 }}
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| ==External links==
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| * [http://nist.gov/dads/HTML/shortestCommonSuperseq.html Dictionary of Algorithms and Data Structures: shortest common supersequence]
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| [[Category:Problems on strings]]
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| [[Category:Combinatorics]]
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| [[Category:Formal languages]]
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| [[Category:Dynamic programming]]
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