In the standard trace reconstruction problem, the goal is to \emph{exactly} reconstruct an unknown source string $\mathsf{x} \in \{0,1\}^n$ from independent "traces", which are copies of $\mathsf{x}$ that have been corrupted by a $\delta$-deletion channel which independently deletes each bit of $\mathsf{x}$ with probability $\delta$ and concatenates the surviving bits. We study the \emph{approximate} trace reconstruction problem, in which the goal is only to obtain a high-accuracy approximation of $\mathsf{x}$ rather than an exact reconstruction. We give an efficient algorithm, and a near-matching lower bound, for approximate reconstruction of a random source string $\mathsf{x} \in \{0,1\}^n$ from few traces. Our main algorithmic result is a polynomial-time algorithm with the following property: for any deletion rate $0 < \delta < 1$ (which may depend on $n$), for almost every source string $\mathsf{x} \in \{0,1\}^n$, given any number $M \leq \Theta(1/\delta)$ of traces from $\mathrm{Del}_\delta(\mathsf{x})$, the algorithm constructs a hypothesis string $\widehat{\mathsf{x}}$ that has edit distance at most $n \cdot (\delta M)^{\Omega(M)}$ from $\mathsf{x}$. We also prove a near-matching information-theoretic lower bound showing that given $M \leq \Theta(1/\delta)$ traces from $\mathrm{Del}_\delta(\mathsf{x})$ for a random $n$-bit string $\mathsf{x}$, the smallest possible expected edit distance that any algorithm can achieve, regardless of its running time, is $n \cdot (\delta M)^{O(M)}$.
翻译:在标准的跟踪重建问题中,目标是重建一个未知的源字符串$\mathfsf{x} 。 我们研究一个未知的源字符串$\mathfs{x} 从独立的“tratchs” $ 0.1\n美元, 美元被一个 $delta$ =masf{x} 美元腐烂的频道, 该频道可以独立删除每位$\mathfsf{x} 美元, 概率为 $=dela{x}, 并连接存不到的位数 。 我们研究的直径{emph{s} 跟踪重建问题, 其中的目标只是获取一个 $\mathfs} $ 0.1xx美元的高精确度近似值的近似值近似值 。 我们的主要算法结果是一个包含以下属性的极值 : 对于任何删除率 $ $ 美元, delx\\\\\\\\\\\\ 美元 ma} 直系它的任何源值 $ 。