Computing and Enumerating Minimal Common Supersequences Between Two Strings

2026-03-23

AI Summary

This work addresses the efficient computation and complete enumeration of all minimal common supersequences (MCS) of two strings, moving beyond the traditional focus on only the shortest common supersequence. The authors propose an algorithmic framework that combines dynamic programming with novel data structures to construct any MCS in O(n) time. Furthermore, after O(n³) preprocessing and using O(n²) space, the framework enables the enumeration of all MCSs with O(n) delay per output sequence. This approach represents the first method capable of generating a single MCS in linear time while supporting full enumeration with controllable delay, significantly enhancing the scalability and practical applicability of MCS computation.

Technology Category

Machine Learning: Semi-Supervised LearningSearch and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsSearch and Retrieval-Augmented AI: Efficiency and scalability of Web search enginesWeb Mining and Content Analysis: Models for Web evolution

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.

Problem

• minimal common supersequence
• string algorithms
• enumeration
• computational complexity

Innovation

• minimal common supersequence
• linear-time algorithm
• enumeration with delay
• string algorithms
• data structure