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.

翻译：在迹重构问题中，我们的目标是通过给定未知字符串 $x\in \{0,1\}^n$ 的独立迹来学习该字符串。一条迹是通过以概率 $\delta$ 独立删除 $x$ 的每一位，并将剩余位连接起来而获得的。当删除概率 $\delta$ 为常数时，迹重构问题是否能在多项式数量的迹下解决，是一个主要的开放性问题。目前已知的最佳上界和下界分别为 $\exp(\tilde O(n^{1/5}))$ 和 $\tilde \Omega(n^{3/2})$，均由 Chase [Cha21b,Cha21a] 给出。我们的主要结果是：如果字符串 $x$ 是轻度分隔的，即 $x$ 中任意两个 1 之间的 0 的数量至少为多对数 $n$，并且如果 $\delta$ 是一个足够小的常数，那么迹重构问题可以用 $O(n \log n)$ 条迹并在多项式时间内解决。