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Computing and Enumerating Minimal Common Supersequences Between Two Strings

Braeden Sopp, Adiesha Liyanage, Mingyang Gong, Binhai Zhu

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.

Computing and Enumerating Minimal Common Supersequences Between Two Strings

Abstract

Given strings each of length at most , computing the shortest common supersequence of them is a well-known NP-hard problem (when is unbounded). On the other hand, when , 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 input strings. Our results are summarized as follows. A minimal common supersequence of input strings can be computed in time. (The method also works when is a constant). All minimal common supersequences between two input strings can be enumerated with a data structure of space and an time delay, and the data structure can be constructed in time.
Paper Structure (11 sections, 21 theorems, 12 equations, 2 figures, 2 algorithms)

This paper contains 11 sections, 21 theorems, 12 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of Minimal Common Supersequences
  4. An $O(n)$ Time Algorithm for Computing an MCS of Two Strings
  5. Enumeration of Minimal Common Supersequences
  6. Graph for Enumeration
  7. Speeding Up the Construction of $G_{st}(A,B)$
  8. An $O(kn\log n )$ Time Algorithm for Computing an MCS of $k$ Strings
  9. A Data Structure for Searching for Occurrences
  10. Reducing a Common Supersequence for $k$ Strings
  11. Concluding Remarks

Key Result

Lemma 2.1

Given strings $A, B$, let $i \in [1:|S|]$, we define $S' = S[1:i)\circ S(i:|S|]$. We say that $S$ is a minimal common supersequence of $A$ and $B$ if and only if $\forall i \in [1:|S|]: S'(i) \text{~is not a common supersequence of } A \text{~and~}B.$

Figures (2)

  • Figure 1: Depicted on the left is $G(A, B)$ containing all nodes with the vertices of $G_{st}(A,B)$ colored in terms of their respective partitions. Red vertices belong partition $V_A$ and blue vertices belong to partition $V_B$. On the right we have $G_{st}(A,B)$ with the vertices labeled. $A=bacba, B=abcca$.
  • Figure 2: On the left hand side, we show characters used in a right embedding of $A_1 =abbc$ into $S=abccbacc$ in orange cells. On the right, we depict the output data structure for $S$ and $A_1$. On the bottom is the result of $\textsc{MergeRightEmbedding}(S;A_1,A_2)$ where $A_2=ac$.

Theorems & Definitions (26)

  • Definition 2.1: $\mathop{\mathrm{\mathbf{MCSupEnum}}}\nolimits$
  • Lemma 2.1
  • Definition 2.2: Embedding of a string
  • Definition 2.3: Left (Right) Embedding
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.1
  • ...and 16 more