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DOI: 10.4230/LIPIcs.ICALP.2023.102
URN: urn:nbn:de:0030-drops-181542
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18154/
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Rubinstein, Ittai

Average-Case to (Shifted) Worst-Case Reduction for the Trace Reconstruction Problem

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LIPIcs-ICALP-2023-102.pdf (0.9 MB)

Abstract

In the trace reconstruction problem, one is given many outputs (called traces) of a noise channel applied to the same input message x, and is asked to recover the input message. Common noise channels studied in the context of trace reconstruction include the deletion channel which deletes each bit w.p. δ, the insertion channel which inserts a G_j i.i.d. uniformly distributed bits before each bit of the input message (where G_j is i.i.d. geometrically distributed with parameter σ) and the symmetry channel which flips each bit of the input message i.i.d. w.p. γ.
De et al. and Nazarov and Peres [De et al., 2017; Nazarov and Peres, 2017] showed that any string x can be reconstructed from exp(O(n^{1/3})) traces. Holden et al. [Holden et al., 2018] adapted the techniques used to prove this upper bound, to construct an algorithm for average-case trace reconstruction from the insertion-deletion channel with a sample complexity of exp(O(log^{1/3} n)). However, it is not clear how to apply their techniques more generally and in particular for the recent worst-case upper bound of exp(Õ(n^{1/5})) shown by Chase [Chase, 2021] for the deletion channel.
We prove a general reduction from the average-case to smaller instances of a problem similar to worst-case and extend Chase’s upper-bound to this problem and to symmetry and insertion channels as well. Using this reduction and generalization of Chase’s bound, we introduce an algorithm for the average-case trace reconstruction from the symmetry-insertion-deletion channel with a sample complexity of exp(Õ(log^{1/5} n)).

BibTeX - Entry

@InProceedings{rubinstein:LIPIcs.ICALP.2023.102,
  author =	{Rubinstein, Ittai},
  title =	{{Average-Case to (Shifted) Worst-Case Reduction for the Trace Reconstruction Problem}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{102:1--102:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18154},
  URN =		{urn:nbn:de:0030-drops-181542},
  doi =		{10.4230/LIPIcs.ICALP.2023.102},
  annote =	{Keywords: Trace Reconstruction, Synchronization Channels, Computational Learning Theory, Computational Biology}
}

Keywords: Trace Reconstruction, Synchronization Channels, Computational Learning Theory, Computational Biology
Collection: 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Issue Date: 2023
Date of publication: 05.07.2023

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