On the Sequence Reconstruction Problem for the Single-Deletion Two-Substitution Channel

2026-01-12

AI Summary

This work investigates the minimum number of noisy copies required to reconstruct an original $q$-ary sequence transmitted over a mixed error channel that introduces a single deletion and up to two substitutions. By leveraging combinatorial and coding-theoretic techniques, the study establishes, for the first time, a tight upper bound on the size of the intersection of error balls under this hybrid channel model. The main contribution is the proof that when the Hamming distance between sequences is at least 2, the intersection size is bounded above by $(q^2 - 1)n^2 - (3q^2 + 5q - 5)n + O_q(1)$, and this bound is tight up to an additive constant. This result fills a critical gap in the theory of sequence reconstruction under multi-type error channels.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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

The Levenshtein sequence reconstruction problem studies the reconstruction of a transmitted sequence from multiple erroneous copies of it. A fundamental question in this field is to determine the minimum number of erroneous copies required to guarantee correct reconstruction of the original sequence. This problem is equivalent to determining the maximum possible intersection size of two error balls associated with the underlying channel. Existing research on the sequence reconstruction problem has largely focused on channels with a single type of error, such as insertions, deletions, or substitutions alone. However, relatively little is known for channels that involve a mixture of error types, for instance, channels allowing both deletions and substitutions. In this work, we study the sequence reconstruction problem for the single-deletion two-substitution channel, which allows one deletion and at most two substitutions applied to the transmitted sequence. Specifically, we prove that if two $q$-ary length-$n$ sequences have the Hamming distance $d\geq 2$, where $q\geq 2$ is any fixed integer, then the intersection size of their error balls under the single-deletion two-substitution channel is upper bounded by $(q^2-1)n^2-(3q^2+5q-5)n+O_q(1)$, where $O_q(1)$ is a constant independent from $n$ but dependent on $q$. Moreover, we show that this upper bound is tight up to an additive constant.

Problem

• sequence reconstruction
• single-deletion two-substitution channel
• error balls
• intersection size
• Levenshtein reconstruction

Innovation

• sequence reconstruction
• single-deletion two-substitution channel
• error ball intersection
• Levenshtein distance
• q-ary sequences