We consider a search problem on trees in which the goal is to find an adversarially placed treasure, while relying on local, partial information. Specifically, each node in the tree holds a pointer to one of its neighbors, termed \emph{advice}. A node is faulty with probability $q$. The advice at a non-faulty node points to the neighbor that is closer to the treasure, and the advice at a faulty node points to a uniformly random neighbor. Crucially, the advice is {\em permanent}, in the sense that querying the same node again would yield the same answer. Let $\Delta$ denote the maximal degree. Roughly speaking, when considering the expected number of {\em moves}, i.e., edge traversals, we show that a phase transition occurs when the {\em noise parameter} $q$ is about $1/\sqrt{\Delta}$. Below the threshold, there exists an algorithm with expected move complexity $O(D\sqrt{\Delta})$, where $D$ is the depth of the treasure, whereas above the threshold, every search algorithm has expected number of moves which is both exponential in $D$ and polynomial in the number of nodes~$n$. In contrast, if we require to find the treasure with probability at least $1-\delta$, then for every fixed $\varepsilon > 0$, if $q<1/\Delta^{\varepsilon}$ then there exists a search strategy that with probability $1-\delta$ finds the treasure using $(\delta^{-1}D)^{O(\frac 1 \varepsilon)}$ moves. Moreover, we show that $(\delta^{-1}D)^{\Omega(\frac 1 \varepsilon)}$ moves are necessary. Besides the number of moves, we also study the number of advice {\em queries} required to find the treasure. Roughly speaking, for this complexity, we show similar threshold results to those previously stated, where the parameter $D$ is replaced by $\log n$.

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用永久嘈杂的建议搜索树：行走和查询算法

我们考虑树上的搜索问题，其目标是在依赖于局部、部分信息的同时找到对抗性放置的宝藏。具体来说，树中的每个节点都持有一个指向其邻居之一的指针，称为 \emph{advice}。一个节点出现故障的概率为 $q$。非故障节点的建议指向离宝藏较近的邻居，故障节点的建议指向均匀随机的邻居。至关重要的是，建议是 {\em Permanent}，因为再次查询同一个节点会产生相同的答案。让 $\Delta$ 表示最大程度。粗略地说，当考虑{\em 移动}的预期次数，即边缘遍历时，我们表明当{\em 噪声参数} $q$ 约为 $1/\sqrt{\Delta}$ 时会发生相变。低于门槛，存在一个具有预期移动复杂度 $O(D\sqrt{\Delta})$ 的算法，其中 $D$ 是宝藏的深度，而在阈值之上，每个搜索算法都有预期的移动次数，在$D$ 和节点数的多项式~$n$。相反，如果我们要求以至少 $1-\delta$ 的概率找到宝藏，那么对于每一个固定的 $\varepsilon > 0$，如果 $q<1/\Delta^{\varepsilon}$ 则存在一个搜索使用 $(\delta^{-1}D)^{O(\frac 1 \varepsilon)}$ 以概率 $1-\delta$ 找到宝藏的策略。此外，我们表明 $(\delta^{-1}D)^{\Omega(\frac 1 \varepsilon)}$ 移动是必要的。除了移动的次数，我们还研究了寻找宝藏所需的建议数{\em 查询}。粗略地说，对于这种复杂性，