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Trace Reconstruction from Local Statistical Queries

Authors Xi Chen , Anindya De , Chin Ho Lee , Rocco A. Servedio

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Author Details

Xi Chen
  • Columbia University, New York, NY, USA
Anindya De
  • University of Pennsylvania, Philadelphia, PA, USA
Chin Ho Lee
  • North Carolina State University, Raleigh, NC, USA
Rocco A. Servedio
  • Columbia University, New York, NY, USA

Funding

  • Chen, Xi: Supported by NSF grants CCF-1703925, IIS-1838154, CCF-2106429 and CCF-2107187.
  • De, Anindya: Supported by NSF grants CCF-1926872, CCF-1910534 and CCF-2045128.
  • Lee, Chin Ho: Supported by Madhu Sudan’s and Salil Vadhan’s Simons Investigator Awards while at Harvard University.
  • Servedio, Rocco A.: Supported by NSF grants IIS-1838154, CCF-2106429, CCF-2211238 and by the Simons Collaboration on Algorithms and Geometry.

Cite AsGet BibTex

Xi Chen, Anindya De, Chin Ho Lee, and Rocco A. Servedio. Trace Reconstruction from Local Statistical Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 52:1-52:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.52

Abstract

The goal of trace reconstruction is to reconstruct an unknown n-bit string x given only independent random traces of x, where a random trace of x is obtained by passing x through a deletion channel. A Statistical Query (SQ) algorithm for trace reconstruction is an algorithm which can only access statistical information about the distribution of random traces of x rather than individual traces themselves. Such an algorithm is said to be 𝓁-local if each of its statistical queries corresponds to an 𝓁-junta function over some block of 𝓁 consecutive bits in the trace. Since several - but not all - known algorithms for trace reconstruction fall under the local statistical query paradigm, it is interesting to understand the abilities and limitations of local SQ algorithms for trace reconstruction. In this paper we establish nearly-matching upper and lower bounds on local Statistical Query algorithms for both worst-case and average-case trace reconstruction. For the worst-case problem, we show that there is an Õ(n^{1/5})-local SQ algorithm that makes all its queries with tolerance τ ≥ 2^{-Õ(n^{1/5})}, and also that any Õ(n^{1/5})-local SQ algorithm must make some query with tolerance τ ≤ 2^{-Ω̃(n^{1/5})}. For the average-case problem, we show that there is an O(log n)-local SQ algorithm that makes all its queries with tolerance τ ≥ 1/poly(n), and also that any O(log n)-local SQ algorithm must make some query with tolerance τ ≤ 1/poly(n).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic inference problems
Keywords
  • trace reconstruction
  • statistical queries
  • algorithmic statistics

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