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arXiv · 2411.18765

Near-Optimal Trace Reconstruction for Mildly Separated Strings

Anders Aamand·Allen Liu·Shyam Narayanan
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Abstract

In the trace reconstruction problem our goal is to learn an unknown string $x\in \{0,1\}^n$ given independent traces of $x$. A trace is obtained by independently deleting each bit of $x$ with some probability $\delta$ and concatenating the remaining bits. It is a major open question whether the trace reconstruction problem can be solved with a polynomial number of traces when the deletion probability $\delta$ is constant. The best known upper bound and lower bounds are respectively $\exp(\tilde O(n^{1/5}))$ and $\tilde \Omega(n^{3/2})$ both by Chase [Cha21b,Cha21a]. Our main result is that if the string $x$ is mildly separated, meaning that the number of zeros between any two ones in $x$ is at least polylog$n$, and if $\delta$ is a sufficiently small constant, then the trace reconstruction problem can be solved with $O(n \log n)$ traces and in polynomial time.

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Anders Aamand, Allen Liu, Shyam Narayanan. 2024-11-27. Near-Optimal Trace Reconstruction for Mildly Separated Strings. https://arxiv.org/abs/2411.18765

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