Some New Results on Sequence Reconstruction Problem for Deletion Channels

2026-01-10

arXiv.org

AI Summary

This study addresses the problem of determining the maximum size $ N(n,3,t) $ of the intersection of two metric balls of radius $ t $ in the deletion channel, whose centers are at least distance 3 apart. Through combinatorial analysis and constructive proofs, the authors establish a general lower bound for $ N(n,3,t) $ when $ n \geq 13 $ and $ t \geq 4 $. Notably, they prove for the first time that this bound is tight when $ t = 4 $, thereby exactly characterizing $ N(n,3,4) = 20n - 166 $. This result resolves an open problem posed by Pham, Goyal, and Kiah, providing a closed-form solution for a critical parameter regime in sequence reconstruction theory.

Technology Category

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

Application Category

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

Abstract

Levenshtein first introduced the sequence reconstruction problem in $2001$. In the realm of combinatorics, the sequence reconstruction problem is equivalent to determining the value of $N(n,d,t)$, which represents the maximum size of the intersection of two metric balls of radius $t$, given that the distance between their centers is at least $d$ and the sequence length is $n$. In this paper, We present a lower bound on $N(n,3,t)$ for $n\geq 13$ and $t \geq 4$. For $t=4$, we prove that this lower bound is tight. This settles an open question posed by Pham, Goyal, and Kiah, confirming that $N(n,3,4)=20n-166$ for all $n \geq 13$.

Problem

• sequence reconstruction
• deletion channels
• metric balls
• combinatorics
• intersection size

Innovation

• sequence reconstruction
• deletion channels
• metric balls
• combinatorial bounds
• Levenshtein problem