### Abstract

In the (deletion-channel) trace reconstruction problem, there is an unknown n-bit source string x. An algorithm is given access to independent traces of x, where a trace is formed by deleting each bit of x independently with probability δ. The goal of the algorithm is to recover x exactly (with high probability), while minimizing samples (number of traces) and running time. Previously, the best known algorithm for the trace reconstruc tion problem was due to Holenstein et al. [SODA 2008]; it uses exp(Õ(n^{1/2})) samples and running time for any fixed 0 < δ < 1. It is also what we call a "mean-based algorithm", meaning that it only uses the empirical means of the individual bits of the traces. Holenstein et al. also gave a lower bound, showing that any mean-based algorithm must use at least n^{Ω(log n}) samples. In this paper we improve both of these results, obtaining match ing upper and lower bounds for mean-based trace reconstruction For any constant deletion rate 0 < δ < 1, we give a mean-based algorithm that uses exp(O(n^{1/3})) time and traces; we also prove that any mean-based algorithm must use at least exp(Ω(n^{1/3})) traces. In fact, we obtain matching upper and lower bounds even for δ subconstant and ρ:= 1 - δ subconstant: when (log^{3} n)/n << δ < 1/2 the bound is exp(-Θ(δn)^{1/3}), and when 1/√n << ρ ≤ 1/2 the bound is exp(-Θ(n/ρ)^{1/3}). Our proofs involve estimates for the maxima of Littlewood polynomials on complex disks. We show that these techniques can also be used to perform trace reconstruction with random insertions and bit-flips in addition to deletions. We also find a surprising result: for deletion probabilities δ > 1/2, the presence of insertions can actually help with trace reconstruction.

Original language | English (US) |
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Title of host publication | STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Pierre McKenzie, Valerie King, Hamed Hatami |

Publisher | Association for Computing Machinery |

Pages | 1047-1056 |

Number of pages | 10 |

ISBN (Electronic) | 9781450345286 |

DOIs | |

State | Published - Jun 19 2017 |

Event | 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 - Montreal, Canada Duration: Jun 19 2017 → Jun 23 2017 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | Part F128415 |

ISSN (Print) | 0737-8017 |

### Other

Other | 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 |
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Country | Canada |

City | Montreal |

Period | 6/19/17 → 6/23/17 |

### Keywords

- Deletion channel
- Littlewood polynomials
- Trace reconstruction

### ASJC Scopus subject areas

- Software

## Cite this

*STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing*(pp. 1047-1056). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. Part F128415). Association for Computing Machinery. https://doi.org/10.1145/3055399.3055450