New Bounds for Circular Trace Reconstruction

2025-12-02

AI Summary

This paper studies trace reconstruction over the cyclic deletion channel: an unknown binary string (x) is observed through traces generated by random bit deletions followed by independent uniform cyclic shifts. The central objective is to determine the minimum number of traces required to reconstruct sparse strings containing a constant number of 1s. Leveraging Fourier analysis, probabilistic modeling, and combinatorial constructions, the authors characterize statistical indistinguishability among traces. Their main contributions are two-fold: (1) they improve the lower bound for cyclic trace reconstruction from (widetilde{Omega}(n^3)) to (widetilde{Omega}(n^5)); and (2) they present the first efficient reconstruction algorithm for constant-sparse strings, requiring only (widetilde{O}(n^6)) traces—exponentially fewer than the best-known exponential upper bound for general strings—thereby revealing the critical role of sparsity in reducing algorithmic complexity.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• 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

The''trace reconstruction''problem asks, given an unknown binary string $x$ and a channel that repeatedly returns''traces''of $x$ with each bit randomly deleted with some probability $p$, how many traces are needed to recover $x$? There is an exponential gap between the best known upper and lower bounds for this problem. Many variants of the model have been introduced in hopes of motivating or revealing new approaches to narrow this gap. We study the variant of circular trace reconstruction introduced by Narayanan and Ren (ITCS 2021), in which traces undergo a random cyclic shift in addition to random deletions. We show an improved lower bound of $ ilde{Omega}(n^5)$ for circular trace reconstruction. This contrasts with the (previously) best known lower bounds of $ ilde{Omega}(n^3)$ in the circular case and $ ilde{Omega}(n^{3/2})$ in the linear case. Our bound shows the indistinguishability of traces from two sparse strings $x,y$ that each have a constant number of nonzeros. Can this technique be extended significantly? How hard is it to reconstruct a sparse string $x$ under a cyclic deletion channel? We resolve these questions by showing, using Fourier techniques, that $ ilde{O}(n^6)$ traces suffice for reconstructing any constant-sparse string in a circular deletion channel, in contrast to the upper bound of $exp( ilde{O}(n^{1/3}))$ for general strings in the circular deletion channel. This shows that new algorithms or new lower bounds must focus on non-constant-sparse strings.

Problem

• Improved lower bound for circular trace reconstruction
• Upper bound for constant-sparse strings in circular deletion channel
• Focus on non-constant-sparse strings for future bounds

Innovation

• Improved lower bound of Ω(n^5) for circular trace reconstruction
• Fourier techniques enable Õ(n^6) traces for constant-sparse strings
• Focus on non-constant-sparse strings for future algorithms or bounds