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Near-Optimal Trace Reconstruction for Mildly Separated Strings

Authors Anders Aamand , Allen Liu , Shyam Narayanan

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  • Theory of computation → Design and analysis of algorithms
Keywords
  • Trace Reconstruction
  • deletion channel
  • sample complexity

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Abstract

In the trace reconstruction problem our goal is to learn an unknown string x ∈ {0,1}ⁿ given independent traces of x. A trace is obtained by independently deleting each bit of x with some probability δ and concatenating the remaining bits. It is a major open question whether the trace reconstruction problem can be solved with a polynomial number of traces when the deletion probability δ is constant. The best known upper bound and lower bounds are respectively exp(Õ(n^{1/5})) [Zachary Chase, 2021a] and ̃ Ω(n^{3/2}) [Zachary Chase, 2021b]. Our main result is that if the string x is mildly separated, meaning that the number of zeros between any two ones in x is at least polylog n, and if δ is a sufficiently small constant, then the trace reconstruction problem can be solved with O(n log n) traces and in polynomial time.

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Anders Aamand, Allen Liu, and Shyam Narayanan. Near-Optimal Trace Reconstruction for Mildly Separated Strings. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.3

Author Details

Anders Aamand
  • BARC, University of Copenhagen, Denmark
Allen Liu
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Shyam Narayanan
  • Citadel Securities, Miami, FL, USA

Funding

  • Aamand, Anders: This work was supported by the DFF-International Postdoc Grant 0164-00022B and by the VILLUM Foundation grants 54451 and 16582.
  • Liu, Allen: This work was supported in part by an NSF Graduate Research Fellowship, a Hertz Fellowship, and a Citadel GQS Fellowship.
  • Narayanan, Shyam: Supported by the NSF TRIPODS Program, an NSF Graduate Fellowship, and a Google Fellowship.

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