Affiliation:
1. CNRS and University of Paris, France
2. Weizmann Institute of Science, Rehovot, Israel
3. Ort Braude College, Israel
Abstract
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.
Funder
Israel Science Foundation
European Research Council
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference32 articles.
1. Optimal Search in Trees
2. Lucas Boczkowski Amos Korman and Yoav Rodeh. 2016. Searching on trees with noisy memory. Retrieved from http://arxiv.org/abs/1611.01403. Lucas Boczkowski Amos Korman and Yoav Rodeh. 2016. Searching on trees with noisy memory. Retrieved from http://arxiv.org/abs/1611.01403.
Cited by
2 articles.
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