Optimal mean-based algorithms for trace reconstruction1,2

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

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


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 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(O(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( 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 < δ < 1, we give a mean-based algorithm that uses exp(O(n1/3)) time and traces; we also prove that any mean-based algorithm must use at least exp((n1/3)) traces. In fact, we obtain matching upper and lower bounds even for δ subconstant and ρ = 1 − δ subconstant: when (log3 n)/n δ ≤ 1/2 the bound is exp(−(δn)1/3), and when 1/n ρ ≤ 1/2 the bound is exp(−(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 languageEnglish (US)
Pages (from-to)851-874
Number of pages24
JournalAnnals of Applied Probability
Issue number2
StatePublished - Apr 2019


  • Deletion channel
  • Littlewood polynomials
  • Trace reconstruction

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


Dive into the research topics of 'Optimal mean-based algorithms for trace reconstruction1,2'. Together they form a unique fingerprint.

Cite this