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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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  • Mathematics of computing → Probabilistic inference problems
Keywords
  • trace reconstruction

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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.

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

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.

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