Computing and Enumerating Minimal Common Supersequences Between Two Strings

TL;DR

This paper presents efficient algorithms for computing and enumerating all minimal common supersequences of two strings, achieving linear time for computation and polynomial space for enumeration.

Contribution

It introduces a linear-time algorithm for finding a minimal common supersequence and a method to enumerate all such sequences with polynomial space and polynomial preprocessing.

Findings

Minimal common supersequence can be computed in O(n) time for two strings.

All minimal common supersequences can be enumerated with O(n^2) space and O(n) delay.

Data structure for enumeration can be built in O(n^3) time.

Abstract

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.