We consider a search problem on trees in which an agent starts at the root of a tree and aims to locate an adversarially placed treasure, by moving along the edges, while relying on local, partial information. Specifically, each node in the tree holds a pointer to one of its neighbors, termed 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 permanent , in the sense that querying the same node again would yield the same answer. Let Δ denote the maximum degree. For the expected number of moves (edge traversals) until finding the treasure, we show that a phase transition occurs when the noise parameter q is roughly 1 √Δ. Below the threshold, there exists an algorithm with expected number of moves O ( D √Δ), where D is the depth of the treasure, whereas above the threshold, every search algorithm has an 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 − δ, then for every fixed ɛ > 0, if q < 1/Δ ɛ , then there exists a search strategy that with probability 1 − δ finds the treasure using (Δ −1 D ) O (1/ε) moves. Moreover, we show that (Δ −1 D ) Ω(1/ε) moves are necessary.

中文翻译：

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使用永久嘈杂的建议在树中导航

我们考虑一个树上的搜索问题，其中代理从树的根部开始，旨在通过沿边缘移动来定位敌对放置的宝藏，同时依赖本地的部分信息。具体来说，树中的每个节点都有一个指向其邻居之一的指针，称为建议. 节点故障概率q. 非故障节点的通知指向距离宝藏较近的邻居，故障节点的建议指向均匀随机的邻居。至关重要的是，建议是永恒的，从某种意义上说，再次查询同一个节点会产生相同的答案。令Δ表示最大度数。对于直到找到宝藏的预期移动次数（边缘遍历），我们表明当噪声参数 q大约为 1 √Δ。低于阈值，存在具有预期移动次数的算法○(D√Δ), 其中D是宝藏的深度，而在阈值之上，每个搜索算法都有一个预期的移动次数，这都是指数D和节点数的多项式n. 相反，如果我们要求找到宝藏的概率至少为 1 − δ，那么对于每个固定的 ɛ > 0，如果q< 1/Δε，则存在一个搜索策略，以概率 1 - δ 找到宝藏，使用 (Δ-1 D) ○(1/ε)移动。此外，我们表明（Δ-1 D)Ω(1/ε)移动是必要的。