Navigating in Trees with Permanently Noisy Advice

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/Δɛ, then there exists a search strategy that with probability 1 − δ finds the treasure using (Δ −1D)O(1/ε) moves. Moreover, we show that (Δ −1D)Ω(1/ε) moves are necessary.