In the trace reconstruction problem, one observes the output of passing a binary string $s \in \{0,1\}^n$ through a deletion channel $T$ times and wishes to recover $s$ from the resulting $T$ "traces." Most of the literature has focused on characterizing the hardness of this problem in terms of the number of traces $T$ needed for perfect reconstruction either in the worst case or in the average case (over input sequences $s$). In this paper, we propose an alternative, instance-based approach to the problem. We define the "Levenshtein difficulty" of a problem instance $(s,T)$ as the probability that the resulting traces do not provide enough information for correct recovery with full certainty. One can then try to characterize, for a specific $s$, how $T$ needs to scale in order for the Levenshtein difficulty to go to zero, and seek reconstruction algorithms that match this scaling for each $s$. For a class of binary strings with alternating long runs, we precisely characterize the scaling of $T$ for which the Levenshtein difficulty goes to zero. For this class, we also prove that a simple "Las Vegas algorithm" has an error probability that decays to zero with the same rate as that with which the Levenshtein difficulty tends to zero.

翻译：在迹线重建问题中，观察者通过删除信道 $T$ 次传递二进制字符串 $s \in \{0,1\}^n$ 的输出，并希望从得到的 $T$ 条"迹线"中恢复 $s$。现有文献主要关注在最坏情况或平均情况（针对输入序列 $s$）下，完美重建所需的迹线数量 $T$ 来刻画该问题的难度。本文提出一种替代性的基于实例的方法来处理该问题。我们将问题实例 $(s,T)$ 的"Levenshtein 难度"定义为：所得迹线无法提供足够信息以确保完全正确恢复的概率。随后可针对特定 $s$，刻画为使 Levenshtein 难度趋于零所需的 $T$ 缩放比例，并寻求对每个 $s$ 均能匹配该缩放比例的重建算法。对于一类具有交替长程结构的二进制字符串，我们精确刻画了使 Levenshtein 难度趋于零所需的 $T$ 缩放比例。针对该类字符串，我们还证明了一种简单的"拉斯维加斯算法"的错误概率以与 Levenshtein 难度趋于零相同的速率衰减至零。