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On the Number of Subsequences in the Nonbinary Deletion Channel

Han Li, Xiang Wang, Fang-Wei Fu

TL;DR

This work analyzes the number of $t$-deletion subsequences $ig|D_t(X)ig|$ in $q$-ary strings under the deletion channel, showing a strong dependence on the run count $r(X)$. It develops a two-pronged approach: a general lower bound achieved by reducing to the binary unbalanced case and applying known results, and an upper bound obtained by focusing on balanced strings $B^q_{r,k}$ whose structure maximizes $ig|D_t(X)ig|$ for fixed $n$ and $r$. The authors derive recursive and, in favorable cases, closed-form expressions for $ig|D_t(B_{r,k;q})ig|$, along with a polynomial-time dynamic-programming algorithm to compute these bounds. The results yield a tight characterization for the extremal $r$-run strings when $r|n$, and point to open questions for the case $r mid n$, with potential implications for deletion-channel coding and sequence reconstruction in nonbinary alphabets.

Abstract

In the deletion channel, an important problem is to determine the number of subsequences derived from a string $U$ of length $n$ when subjected to $t$ deletions. It is well-known that the number of subsequences in the setting exhibits a strong dependence on the number of runs in the string $U$, where a run is defined as a maximal substring of identical characters. In this paper we study the number of subsequences of a non-binary string in this scenario, and propose some improved bounds on the number of subsequences of $r$-run non-binary strings. Specifically, we characterize a family of $r$-run non-binary strings with the maximum number of subsequences under any $t$ deletions, and show that this number can be computed in polynomial time.

On the Number of Subsequences in the Nonbinary Deletion Channel

TL;DR

This work analyzes the number of -deletion subsequences in -ary strings under the deletion channel, showing a strong dependence on the run count . It develops a two-pronged approach: a general lower bound achieved by reducing to the binary unbalanced case and applying known results, and an upper bound obtained by focusing on balanced strings whose structure maximizes for fixed and . The authors derive recursive and, in favorable cases, closed-form expressions for , along with a polynomial-time dynamic-programming algorithm to compute these bounds. The results yield a tight characterization for the extremal -run strings when , and point to open questions for the case , with potential implications for deletion-channel coding and sequence reconstruction in nonbinary alphabets.

Abstract

In the deletion channel, an important problem is to determine the number of subsequences derived from a string of length when subjected to deletions. It is well-known that the number of subsequences in the setting exhibits a strong dependence on the number of runs in the string , where a run is defined as a maximal substring of identical characters. In this paper we study the number of subsequences of a non-binary string in this scenario, and propose some improved bounds on the number of subsequences of -run non-binary strings. Specifically, we characterize a family of -run non-binary strings with the maximum number of subsequences under any deletions, and show that this number can be computed in polynomial time.
Paper Structure (11 sections, 23 theorems, 91 equations, 1 figure, 1 table)

This paper contains 11 sections, 23 theorems, 91 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Unbalanced strings and our lower bound
  4. Balanced strings
  5. Cyclic operation
  6. Balancing operation
  7. Balanced strings
  8. Our upper bound
  9. Recursive expression
  10. Solving the Recursion
  11. Conclusion

Key Result

Lemma 1

Let $n,m_1,m_2,q$ be positive integers such that $m_1+m_2\leq n-t$. Then, for any $U\in \Sigma_q^n$ and $q\geqslant2$ we have

Figures (1)

  • Figure 1: Comparison of upper bounds for the case $q=3$, $n=120$, $r=24$. Our upper bounds derived from balanced strings [Corollary \ref{['Corcomputeb']}] are compared against previous best known bounds. [L] marks the upper bound proven by Levenstein L0. [HR] marks the upper bound proven by Hirschberg and Regnier Hirschberg. The results are presented as functions of $t$ on a logarithmic scale.

Theorems & Definitions (47)

  • Lemma 1
  • Lemma 2
  • Lemma 3: Insertion Increases the Number of Subsequences Hirschberg
  • Lemma 4
  • Lemma 5: Permutation Keeps the Number of Subsequences Unchanged
  • proof
  • Lemma 6: Reduction Operation Decreases the Number of Subsequences
  • proof
  • Remark 1
  • Theorem 7
  • ...and 37 more