Abstract
We consider a search problem on trees aiming to find a treasure that an adversary places at one of the nodes. The algorithm can query nodes and extract directional information from them. That is, each node holds a pointer, termed advice, to one of its neighbors. Ideally, this advice points to the neighbor that is closer to the treasure, however, with probability this advice points to a uniformly random neighbor. Crucially, the advice is permanent, hence querying the same node again yields the same answer. Let Δ denote the maximal degree. Roughly speaking, we show that the expected number of queries incurs a phase transition when is about 1/√Δ. In a recent paper, at TALG'21, we showed that if is above the threshold then the expected number of queries is polynomial in . Here, we prove that below the threshold, the expected number of queries is O(√Δ log Δ · log2n), which is tight up to an factor when is small. We further show that this factor can be reduced to in the case of regular trees and assuming that for sufficiently small c > 0. In addition, we study the case that the treasure must be found with some given probability. We show that for every fixed ϵ, δ > 0, if q < 1/Δϵ then there exists a search strategy that with probability 1-δ finds the treasure using (δ-1log n)O(1/ϵ) queries, whereas (δ-1log n)Ω(1/ϵ) queries are necessary.
Original language | English |
---|---|
Article number | 18 |
Number of pages | 30 |
Journal | ACM Transactions on Algorithms |
Volume | 21 |
Issue number | 2 |
DOIs | |
State | Published - 7 Feb 2025 |
Keywords
- Fault-tolerance
- Noisy advice
- Randomized algorithms
- Search algorithms
- Trees
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)