We study the trace reconstruction problem for spider graphs. Let $n$ be the number of nodes of a spider and $d$ be the length of each leg, and suppose that we are given independent traces of the spider from a deletion channel in which each non-root node is deleted with probability $q$. This is a natural generalization of the string trace reconstruction problem in theoretical computer science, which corresponds to the special case where the spider has one leg. In the regime where $d\ge \log_{1/q}(n)$, the problem can be reduced to the vanilla string trace reconstruction problem. We thus study the more interesting regime $d\le \log_{1/q}(n)$, in which entire legs of the spider are deleted with non-negligible probability. We describe an algorithm that reconstructs spiders with high probability using $\exp\left(\mathcal{O}\left(\frac{(nq^d)^{1/3}}{d^{1/3}}(\log n)^{2/3}\right)\right)$ traces. Our algorithm works for all deletion probabilities $q\in(0,1)$.
翻译：我们研究蜘蛛图的微量重建问题。 美元应该是蜘蛛的节点数, 美元应该是每条腿的长度, 假设我们从一个删除的频道中单独获得蜘蛛的痕迹, 其中每个非根节点被删除, 概率为$q美元。 这是理论计算机科学中字符串的微量重建问题的自然概括化, 这与蜘蛛有一条腿的特殊案例相对应。 在这个制度下, 问题可以降低到香草弦的微量重建问题 。 因此, 我们研究更有趣的制度 $d\le\ log\ 1/q}(n), 其中蜘蛛的整腿被删除, 概率不明显。 我们描述一种算法, 利用 $\ excleft (nqcc)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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