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April 2019 Optimal mean-based algorithms for trace reconstruction
Anindya De, Ryan O’Donnell, Rocco A. Servedio
Ann. Appl. Probab. 29(2): 851-874 (April 2019). DOI: 10.1214/18-AAP1394


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.


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Anindya De. Ryan O’Donnell. Rocco A. Servedio. "Optimal mean-based algorithms for trace reconstruction." Ann. Appl. Probab. 29 (2) 851 - 874, April 2019.


Received: 1 September 2017; Revised: 1 April 2018; Published: April 2019
First available in Project Euclid: 24 January 2019

zbMATH: 07047440
MathSciNet: MR3910019
Digital Object Identifier: 10.1214/18-AAP1394

Primary: 62G07, 68Q32
Secondary: 94A40

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.29 • No. 2 • April 2019
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