The {\em insertion-deletion channel} takes as input a binary string $x \in\{0, 1\}^n$, and outputs a string $\widetilde{x}$ where some of the bits have been deleted and others inserted independently at random. In the {\em trace reconstruction problem}, one is given many outputs (called {\em traces}) of the insertion-deletion channel on the same input message $x$, and asked to recover the input message. Nazarov and Peres (STOC 2017), and De, Odonnell and Servido (STOC 2017) showed that any string $x$ can be reconstructed from $\exp(O(n^{1/3}))$ traces. Holden, Pemantle, Peres and Zhai (COLT 2018) adapt the techniques used to prove this upper bound, to an algorithm for the average-case trace reconstruction with a sample complexity of $\exp(O(\log^{1/3} n))$. However, it is not clear how to apply their techniques more generally and in particular for the recent worst-case upper bound of $\exp(\widetilde{O}(n^{1/5}))$ shown by Chase (STOC 2021) for the deletion-channel. We prove a general reduction from the average-case to smaller instances of a problem similar to worst-case. Using this reduction and a generalization of Chase's bound, we construct an improved average-case algorithm with a sample complexity of $\exp(\widetilde{O}(\log^{1/5} n))$. Additionally, we show that Chase's upper-bound holds for the insertion-deletion channel as well.

翻译：{em 插入- 删除通道} 将输入一个二进制字符串 $x $x {in ⁇ 0, 1 ⁇ n$ 和输出 一个字符串 $x $x {x}, 其中部分位数已被删除, 其它部分随机插入 。 在 {em track Reformation 问题} 中, 给一个插入- 删除频道的许多输出( 称为 {em track} $x $x, 并被要求恢复输入信息 。 Nazarov 和 Peres (STOC 2017) 和 De, Odonell 和 Servido (STOC 2017) 显示, 任何字符串 $x 频道都可以从 $\ explace (O\ 1/ 1/ 3} 美元( On) 中重新重建一个字符串 。 Holden, Peres 和 Zhai- developreal exion a develop la develop le) ex (O_ developlevelop la- develople) lax a we develop develop lovelop le) 20_ le) loceal (O_ le) lovelopmental) lod_ loceloceloceal) la lax lax lobal (我们 lex) legal_ 20x) levelopmental_ clobal_ legal_ 。