Complexity
For the general case of an arbitrary number of input sequences, the problem is NP-hard. When the number of sequences is constant, the problem is solvable in polynomial time by dynamic programming (see Solution below). Assume you have sequences of lengths . A naive search would test each of the subsequences of the first sequence to determine whether they are also subsequences of the remaining sequences; each subsequence may be tested in time linear in the lengths of the remaining sequences, so the time for this algorithm would be
For the case of two sequences of n and m elements, the running time of the dynamic programming approach is O(n × m). For an arbitrary number of input sequences, the dynamic programming approach gives a solution in
There exist methods with lower complexity, which often depend on the length of the LCS, the size of the alphabet, or both.
Notice that the LCS is not necessarily unique; for example the LCS of "ABC" and "ACB" is both "AB" and "AC". Indeed the LCS problem is often defined to be finding all common subsequences of a maximum length. This problem inherently has higher complexity, as the number of such subsequences is exponential in the worst case, even for only two input strings.
Read more about this topic: Longest Common Subsequence Problem
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