Near-Optimal Trace Reconstruction for Mildly Separated Strings
Anders Aamand, Allen Liu, Shyam Narayanan
TL;DR
This work tackles trace reconstruction for constant-deletion channels under the mild separation assumption that zeros between consecutive ones in $x$ number at least $L=\Theta((\log n)^8)$. It introduces a novel left-to-right alignment procedure that preserves a monotone alignment of ones and yields tight coarse gap estimates with high probability, followed by a robust forward-backward scheme to upgrade these estimates to exact gap lengths. The main contribution is a polynomial-time algorithm that uses $N=O(n\log n)$ traces to exactly reconstruct $x$ for $L$-separated strings when $\delta$ is a small constant, with a detailed probabilistic analysis ensuring that the alignment is not ahead and that the probability of falling behind is appropriately bounded. This advances the understanding of tractable instances in trace reconstruction, demonstrates near-optimal trace complexity for this class, and provides techniques (alignment plus forward/backward verification) that may inform future approaches to broader classes of strings and deletion probabilities.
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 $δ$ 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 $δ$ is constant. The best known upper bound and lower bounds are respectively $\exp(\tilde O(n^{1/5}))$ and $\tilde Ω(n^{3/2})$ 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 $δ$ is a sufficiently small constant, then the trace reconstruction problem can be solved with $O(n \log n)$ traces and in polynomial time.