Computing and Enumerating Minimal Common Supersequences Between Two Strings

Given $k$ strings each of length at most $n$, computing the shortest common supersequence of them is a well-known NP-hard problem (when $k$ is unbounded). On the other hand, when $k=2$, such a shortest common supersequence can be computed in $O(n^2)$ time using dynamic programming as a textbook example. In this paper, we consider the problem of computing a \emph{minimal} common supersequence and enumerating all minimal common supersequences for $k=2$ input strings. Our results are summarized as follows. A minimal common supersequence of $k=2$ input strings can be computed in $O(n)$ time. (The method also works when $k$ is a constant). All minimal common supersequences between two input strings can be enumerated with a data structure of $O(n^2)$ space and an $O(n)$ time delay, and the data structure can be constructed in $O(n^3)$ time.