On the Number of Subsequences in the Nonbinary Deletion Channel

2026-01-10

arXiv.org

AI Summary

This study addresses the problem of determining a tight upper bound on the number of distinct subsequences obtainable from a non-binary string after t deletions. By analyzing the run-length structure of strings, the authors fully characterize—for the first time—the structure of a string with r runs that maximizes the number of subsequences under any t deletions. Leveraging combinatorial mathematics and run-based modeling, they devise a polynomial-time algorithm that exactly computes this maximum subsequence count. The work not only establishes a significantly tighter upper bound than previously known but also enables efficient computation of both the optimal string structure and its corresponding subsequence count, thereby advancing the theoretical foundations of subsequence complexity in deletion channels.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• deletion channel
• subsequences
• non-binary strings
• runs
• combinatorics

Innovation

• deletion channel
• subsequence enumeration
• non-binary strings
• run structure
• polynomial-time computation