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Enumeration of Maximal Common Subsequences Between Two Strings

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Abstract

A maximal common subsequence (MCS) between two strings X and Y is an inclusion-maximal subsequence of both X and Y. MCSs are a natural generalization of the classical concept of longest common subsequence (LCS), which can be seen as a longest MCS. We study the problem of efficiently listing all the distinct MCSs between two strings. As discussed in the paper, this problem is algorithmically challenging as the same MCS cannot be listed multiple times: for example, dynamic programming [Fraser et al., CPM 1998] incurs in an exponential waste of time, and a recent algorithm for finding an MCS [Sakai, CPM 2018] does not seem to immediately extend to listing. We follow an alternative and novel graph-based approach, proposing the first output-sensitive algorithm for this problem: it takes polynomial time in n per MCS found, where \(n = \max \{ |X|, |Y|\}\), with polynomial preprocessing time and space.

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Notes

  1. The Strong Exponential Time Hypothesis (SETH) [13] states that \(\lim _{k\rightarrow \infty } s_k = 1\), where \(s_k=\inf \{\delta \mid k\text {-SAT can be solved in } O(2^{\delta n}) \text { time}\}\). It is widely believed to be true, and it has been used to prove conditional lower bounds for a variety of problems (see [2] for some examples).

  2. The author employs these edges to improve the Hunt–Szymansky algorithm [12], which extracts one LCS of two strings of length n in \(O((r + n) \log n)\), where r is the total number of ordered pairs of positions at which the two sequences match, that is, the number of edges in the string bipartite graph.

  3. To be precise, each recursive node has up to \(\sigma \) child nodes; the binary partition is seen by the fact that each recursive call corresponds to “using the edge \((l_c,m_c)\)” and the continuation after backtracking corresponds to “not using the edge \((l_c,m_c)\)”.

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Authors and Affiliations

  1. University of Pisa, Pisa, Italy

    Alessio Conte, Roberto Grossi & Giulia Punzi

  2. National Institute of Informatics, Tokyo, Japan

    Takeaki Uno

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  1. Alessio Conte
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  2. Roberto Grossi
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  3. Giulia Punzi
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  4. Takeaki Uno
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Correspondence to Giulia Punzi.

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Conte, A., Grossi, R., Punzi, G. et al. Enumeration of Maximal Common Subsequences Between Two Strings. Algorithmica 84, 757–783 (2022). https://doi.org/10.1007/s00453-021-00898-5

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