# Polynomial-time trace reconstruction in the smoothed complexity model

@inproceedings{Chen2021PolynomialtimeTR, title={Polynomial-time trace reconstruction in the smoothed complexity model}, author={Xianmiao Chen and Anindya De and Chin Ho Lee and R. Servedio and S. Sinha}, booktitle={SODA}, year={2021} }

In the \emph{trace reconstruction problem}, an unknown source string $x \in \{0,1\}^n$ is sent through a probabilistic \emph{deletion channel} which independently deletes each bit with probability $\delta$ and concatenates the surviving bits, yielding a \emph{trace} of $x$. The problem is to reconstruct $x$ given independent traces. This problem has received much attention in recent years both in the worst-case setting where $x$ may be an arbitrary string in $\{0,1\}^n$ \cite{DOS17… Expand

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#### 14 Citations

Near-Optimal Average-Case Approximate Trace Reconstruction from Few Traces

- Computer Science
- ArXiv
- 2021

An efficient algorithm is given, and a near-matching lower bound is proved, for approximate reconstruction of a random source string x ∈ {0, 1}n from few traces and it is proved that given M ≤ Θ(1/δ) traces from Delδ(x) for a random n-bit string x, the smallest possible expected edit distance that any algorithm can achieve, regardless of its running time, is n · (δM). Expand

New Upper Bounds for Trace Reconstruction

- Computer Science, Mathematics
- ArXiv
- 2020

The lower bound on average case trace reconstruction from Omega's log 9/4 n to Omega's n 3/2 n is improved. Expand

Limitations of Mean-Based Algorithms for Trace Reconstruction at Small Distance

- Computer Science, Mathematics
- 2021 IEEE International Symposium on Information Theory (ISIT)
- 2021

A connection to the famous Prouhet-Tarry-Escott (PTE) problem is described, which shows a barrier to finding explicit hard-to-distinguish strings that would imply explicit short solutions to the PTE problem, a well-known difficult problem in number theory. Expand

Approximate Trace Reconstruction via Median String (in Average-Case)

- Computer Science
- ArXiv
- 2021

An approximate version of the trace reconstruction problem, where the goal is to recover an unknown string s ∈ {0, 1} from m traces is considered, and a deterministic near-linear time algorithm for the average-case model that uses only three traces is presented. Expand

Mean-Based Trace Reconstruction over Practically any Replication-Insertion Channel

- Computer Science, Mathematics
- 2021 IEEE International Symposium on Information Theory (ISIT)
- 2021

This work uses a simple extension of the original complex-analytic approach to show that exp(O(n)) traces suffice for mean-based worst-case trace reconstruction over any memoryless channel that maps each input bit to an arbitrarily distributed sequence of replications and insertions of random bits, provided the length of this sequence follows a subexponential distribution. Expand

Approximate trace reconstruction of random strings from a constant number of traces

- Mathematics
- 2021

In the trace reconstruction problem, the goal is to reconstruct an unknown string x of length n from multiple traces obtained by passing x through the deletion channel. In the relaxed problem of… Expand

Tree trace reconstruction using subtraces

- Computer Science, Mathematics
- ArXiv
- 2021

In these proofs, the notion of a subtrace is introduced, which enables us to connect with and generalize recent mean-based complex analytic algorithms for string trace reconstruction. Expand

Trace Reconstruction Problems in Computational Biology

- Medicine, Computer Science
- IEEE Transactions on Information Theory
- 2021

Several new trace generation models and open questions relevant to trace reconstruction for immunogenomics and DNA data storage are introduced, theoretical results on trace reconstruction are surveyed, and their connections to computational biology are surveyed. Expand

Separating words and trace reconstruction

- Computer Science
- STOC
- 2021

It is proved that for any distinct x,y ∈ {0,1}n, there is a deterministic finite automaton with O(n1/3) states that accepts x but not y, and the upper bound on worst case trace reconstruction is improved. Expand

PR ] 7 S ep 2 02 0 NEW UPPER BOUNDS FOR TRACE RECONSTRUCTION

- 2020

We improve the upper bound on worst case trace reconstruction from exp(O(n)) to exp(Õ(n)) for any deletion probability q ≤ 1 2 .

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