The Query Complexity of Searching Trees with Permanently Noisy Advice

Lucas Boczkowski, Uriel Feige, Amos Korman, Yoav Rodeh

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Article number18
Number of pages30
JournalACM Transactions on Algorithms
Volume21
Issue number2
DOIs
StatePublished - 7 Feb 2025

Keywords

  • Fault-tolerance
  • Noisy advice
  • Randomized algorithms
  • Search algorithms
  • Trees

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)

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