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Polynomial-Time Trace Reconstruction in the Low Deletion Rate Regime

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

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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
  • Columbia University, New York, NY, USA
Rocco A. Servedio
  • Columbia University, New York, NY, USA
Sandip Sinha
  • Columbia University, New York, NY, USA

Funding

  • Chen, Xi: Supported by NSF grants CCF-1703925 and IIS-1838154.
  • De, Anindya: Supported by NSF grants CCF-1926872 and CCF-1910534.
  • Lee, Chin Ho: Supported by a grant from the Croucher Foundation and by the Simons Collaboration on Algorithms and Geometry.
  • Servedio, Rocco A.: Supported by NSF grants CCF-1814873, IIS-1838154, CCF-1563155, and by the Simons Collaboration on Algorithms and Geometry.
  • Sinha, Sandip: Supported by NSF grants CCF-1714818, CCF-1822809, IIS-1838154, CCF-1617955, CCF-1740833, and by the Simons Collaboration on Algorithms and Geometry.

Cite As Get BibTex

Xi Chen, Anindya De, Chin Ho Lee, Rocco A. Servedio, and Sandip Sinha. Polynomial-Time Trace Reconstruction in the Low Deletion Rate Regime. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ITCS.2021.20

Abstract

In the trace reconstruction problem, an unknown source string x ∈ {0,1}ⁿ is transmitted through a probabilistic deletion channel which independently deletes each bit with some fixed probability δ and concatenates the surviving bits, resulting in a trace of x. The problem is to reconstruct x given access to independent traces. Trace reconstruction of arbitrary (worst-case) strings is a challenging problem, with the current state of the art for poly(n)-time algorithms being the 2004 algorithm of Batu et al. [T. Batu et al., 2004]. This algorithm can reconstruct an arbitrary source string x ∈ {0,1}ⁿ in poly(n) time provided that the deletion rate δ satisfies δ ≤ n^{-(1/2 + ε)} for some ε > 0.
In this work we improve on the result of [T. Batu et al., 2004] by giving a poly(n)-time algorithm for trace reconstruction for any deletion rate δ ≤ n^{-(1/3 + ε)}. Our algorithm works by alternating an alignment-based procedure, which we show effectively reconstructs portions of the source string that are not "highly repetitive", with a novel procedure that efficiently determines the length of highly repetitive subwords of the source string.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic inference problems
Keywords
  • trace reconstruction

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References

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