The Annals of Applied Probability

Optimal mean-based algorithms for trace reconstruction

Anindya De, Ryan O’Donnell, and Rocco A. Servedio

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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 $\delta$. 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 reconstruction problem was due to Holenstein et al. [in Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms 389–398 (2008) ACM]; it uses $\exp(\widetilde{O}(n^{1/2}))$ samples and running time for any fixed $0<\delta<1$. 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 $n^{\widetilde{\Omega}(\log n)}$ samples.

In this paper, we improve both of these results, obtaining matching upper and lower bounds for mean-based trace reconstruction. For any constant deletion rate $0<\delta<1$, 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 $\exp(\Omega(n^{1/3}))$ traces. In fact, we obtain matching upper and lower bounds even for $\delta$ subconstant and $\rho\:=1-\delta$ subconstant: when $(\log^{3}n)/n\ll\delta\leq1/2$ the bound is $\exp(-\Theta(\delta n)^{1/3})$, and when $1/\sqrt{n}\ll\rho\leq1/2$ the bound is $\exp(-\Theta(n/\rho)^{1/3})$.

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 $\delta>1/2$, the presence of insertions can actually help with trace reconstruction.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 851-874.

Dates
Received: September 2017
Revised: April 2018
First available in Project Euclid: 24 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1548298932

Digital Object Identifier
doi:10.1214/18-AAP1394

Mathematical Reviews number (MathSciNet)
MR3910019

Zentralblatt MATH identifier
07047440

Subjects
Primary: 62G07: Density estimation 68Q32: Computational learning theory [See also 68T05]
Secondary: 94A40: Channel models (including quantum)

Keywords
Trace reconstruction deletion channel Littlewood polynomials

Citation

De, Anindya; O’Donnell, Ryan; Servedio, Rocco A. Optimal mean-based algorithms for trace reconstruction. Ann. Appl. Probab. 29 (2019), no. 2, 851--874. doi:10.1214/18-AAP1394. https://projecteuclid.org/euclid.aoap/1548298932


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