Searching on Trees with Noisy Memory.
Lucas Boczkowski, Amos Korman, Yoav Rodeh · arXiv (Cornell University) · 2016
We consider a search problem on trees using unreliable guiding instructions. Specifically, an agent starts a search at the root of a tree aiming to find a treasure hidden at one of the nodes by an adversary. Each visited node holds information, called advice, regarding the most promising neighbor to continue the search. However, the memory holding this information may be unreliable. Modeling this scenario, we focus on a probabilistic setting. That is, the advice at a node is a pointer to one of its neighbors. With probability $q$ each node is faulty, independently of other nodes, in which case its advice points at an arbitrary neighbor, chosen u.a.r. Otherwise, the node is sound and necessarily points at the correct neighbor. Crucially, the advice is permanent, in the sense that querying a node several times would yield the same answer. We evaluate the agent's efficiency by two measures: The move complexity denotes the expected number of edge traversals, and the query complexity denotes the expected number of queries. Let $\Delta$ denote the maximal degree. Roughly speaking, the main message of this paper is that in order to obtain efficient search, $1/\sqrt{\Delta}$ is a threshold for the noise parameter $q$. Essentially, we prove that above the threshold, every search algorithm has query complexity (and move complexity) which is both exponential in the depth $d$ of the treasure and polynomial in the number of nodes $n$. Conversely, below the threshold, there exists an algorithm with move complexity $O(d\sqrt{\Delta})$, and an algorithm with query complexity $O(\sqrt{\Delta}\log \Delta \log^2 n)$. Moreover, for the case of regular trees, we obtain an algorithm with query complexity $O(\sqrt{\Delta}\log n\log\log n)$. The move complexity bound is tight below the threshold and the query complexity bounds are not far from the lower bound of $\Omega(\sqrt{\Delta}\log_\Delta n)$.