## The Annals of Applied Probability

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

### Optimal mean-based algorithms for trace reconstruction

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

#### 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