The goal of trace reconstruction is to reconstruct an unknown $n$-bit string $x$ given only independent random traces of $x$, where a random trace of $x$ is obtained by passing $x$ through a deletion channel. A Statistical Query (SQ) algorithm for trace reconstruction is an algorithm which can only access statistical information about the distribution of random traces of $x$ rather than individual traces themselves. Such an algorithm is said to be $\ell$-local if each of its statistical queries corresponds to an $\ell$-junta function over some block of $\ell$ consecutive bits in the trace. Since several -- but not all -- known algorithms for trace reconstruction fall under the local statistical query paradigm, it is interesting to understand the abilities and limitations of local SQ algorithms for trace reconstruction. In this paper we establish nearly-matching upper and lower bounds on local Statistical Query algorithms for both worst-case and average-case trace reconstruction. For the worst-case problem, we show that there is an $\tilde{O}(n^{1/5})$-local SQ algorithm that makes all its queries with tolerance $\tau \geq 2^{-\tilde{O}(n^{1/5})}$, and also that any $\tilde{O}(n^{1/5})$-local SQ algorithm must make some query with tolerance $\tau \leq 2^{-\tilde{\Omega}(n^{1/5})}$. For the average-case problem, we show that there is an $O(\log n)$-local SQ algorithm that makes all its queries with tolerance $\tau \geq 1/\mathrm{poly}(n)$, and also that any $O(\log n)$-local SQ algorithm must make some query with tolerance $\tau \leq 1/\mathrm{poly}(n).$

翻译：迹重构的目标是仅通过未知的 $n$ 比特串 $x$ 的独立随机迹来重构该串，其中 $x$ 的随机迹是通过将 $x$ 送入删除信道而获得的。迹重构的统计查询（SQ）算法是一种只能访问 $x$ 的随机迹分布的统计信息，而非单个迹本身的算法。如果该算法的每个统计查询对应于迹中某个连续 $\ell$ 比特块上的 $\ell$-junta 函数，则称该算法是 $\ell$-局部的。由于已知的迹重构算法中，部分（但并非全部）属于局部统计查询范式，因此理解局部 SQ 算法在迹重构中的能力和局限性具有重要意义。本文针对最坏情况和平均情况下的迹重构，建立了局部统计查询算法的近乎匹配的上界和下界。对于最坏情况问题，我们证明存在一个 $\tilde{O}(n^{1/5})$-局部 SQ 算法，其所有查询的容差为 $\tau \geq 2^{-\tilde{O}(n^{1/5})}$，并且任何 $\tilde{O}(n^{1/5})$-局部 SQ 算法都必须进行某些容差 $\tau \leq 2^{-\tilde{\Omega}(n^{1/5})}$ 的查询。对于平均情况问题，我们证明存在一个 $O(\log n)$-局部 SQ 算法，其所有查询的容差为 $\tau \geq 1/\mathrm{poly}(n)$，并且任何 $O(\log n)$-局部 SQ 算法都必须进行某些容差 $\tau \leq 1/\mathrm{poly}(n)$ 的查询。