Searching a Tree with Permanently Noisy Advice

Authors Lucas Boczkowski, Amos Korman, Yoav Rodeh

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Author Details

Lucas Boczkowski
  • CNRS, IRIF, Univ. Paris Diderot, Paris, France
Amos Korman
  • CNRS, IRIF, Univ. Paris Diderot, Paris, France
Yoav Rodeh
  • Dep. of Physics of Complex Systems. Weizmann Institute, Rehovot, Israel

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Lucas Boczkowski, Amos Korman, and Yoav Rodeh. Searching a Tree with Permanently Noisy Advice. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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 uniformly at random. Otherwise, the node is sound and 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 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 a phase transition occurs when the noise parameter q is roughly 1/sqrt{Delta}. More precisely, 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). For q that is below but close to the threshold, the bound for the move complexity is tight, and the bounds for the query complexity are not far from the lower bound of Omega(sqrt{Delta}log_Delta n). In addition, we also consider a semi-adversarial variant, in which an adversary chooses the direction of advice at faulty nodes. For this variant, the threshold for efficient moving algorithms happens when the noise parameter is roughly 1/Delta. Above this threshold a simple protocol that follows each advice with a fixed probability already achieves optimal move complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Database theory
  • Theory of computation → Theory of randomized search heuristics
  • Data structures
  • Graph search
  • Average Case Analysis


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  1. Andrei Asinowski, Jean Cardinal, Nathann Cohen, Sébastien Collette, Thomas Hackl, Michael Hoffmann, Kolja B. Knauer, Stefan Langerman, Michal Lason, Piotr Micek, Günter Rote, and Torsten Ueckerdt. Coloring hypergraphs induced by dynamic point sets and bottomless rectangles. CoRR, abs/1302.2426, 2013. URL:
  2. Javed A. Aslam and Aditi Dhagat. Searching in the presence of linearly bounded errors. In STOC, pages 486-493. ACM, 1991. URL:
  3. Yosi Ben-Asher, Eitan Farchi, and Ilan Newman. Optimal search in trees. SIAM J. Comput., 28(6):2090-2102, 1999. URL:
  4. Michael Ben-Or and Avinatan Hassidim. The bayesian learner is optimal for noisy binary search (and pretty good for quantum as well). In FOCS, pages 221-230, 2008. URL:
  5. Lucas Boczkowski, Amos Korman, and Yoav Rodeh. Searching a tree with permanently noisy advice., 2016. URL:
  6. Ryan S. Borgstrom and S. Rao Kosaraju. Comparison-based search in the presence of errors. In STOC, pages 130-136. ACM, 1993. Google Scholar
  7. Mark Braverman and Elchanan Mossel. Noisy sorting without resampling. In SODA, pages 268-276, 2008. URL:
  8. Gerth Stølting Brodal, Rolf Fagerberg, Irene Finocchi, Fabrizio Grandoni, Giuseppe F. Italiano, Allan Grønlund Jørgensen, Gabriel Moruz, and Thomas Mølhave. Optimal resilient dynamic dictionaries. In Algorithms - ESA 2007, 15th Annual European Symposium, Eilat, Israel, October 8-10, 2007, Proceedings, pages 347-358, 2007. URL:
  9. Ferdinando Cicalese and Ugo Vaccaro. Optimal strategies against a liar. Theor. Comput. Sci., 230(1-2):167-193, 2000. Google Scholar
  10. Alexander Drewitz and Alejandro F. Ramiréz. Selected topics in random walk in random environment. Topics in Percolative and Disordered Systems, Springer Proceedings in Mathematics and Statistics, 69:23-83, 2014. Google Scholar
  11. Ehsan Emamjomeh-Zadeh, David Kempe, and Vikrant Singhal. Deterministic and probabilistic binary search in graphs. In STOC, pages 519-532, 2016. URL:
  12. Uriel Feige, Prabhakar Raghavan, David Peleg, and Eli Upfal. Computing with noisy information. SIAM J. Comput., 23(5):1001-1018, 1994. URL:
  13. Irene Finocchi, Fabrizio Grandoni, and Giuseppe F. Italiano. Resilient search trees. In SODA, pages 547-553, 2007. URL:
  14. Irene Finocchi and Giuseppe F. Italiano. Sorting and searching in the presence of memory faults (without redundancy). In STOC, pages 101-110, 2004. URL:
  15. Ehud Fonio, Yael Heyman, Lucas Boczkowski, Aviram Gelblum, Adrian Kosowski, Amos Korman, and Ofer Feinerman. A locally-blazed ant trail achieves efficient collective navigation despite limited information, eLife 2016;5:e20185, 2016. URL:
  16. Nicolas Hanusse, David Ilcinkas, Adrian Kosowski, and Nicolas Nisse. Locating a target with an agent guided by unreliable local advice: How to beat the random walk when you have a clock? In PODC, pages 355-364, 2010. URL:
  17. Nicolas Hanusse, Dimitris Kavvadias, Evangelos Kranakis, and Danny Krizanc. Memoryless search algorithms in a network with faulty advice. Theoretical Computer Science, 402(2–3):190-198, 2008. URL:
  18. Nicolas Hanusse, Evangelos Kranakis, and Danny Krizanc. Searching with mobile agents in networks with liars. Discrete Applied Mathematics, 137(1):69-85, 2004. URL:
  19. Richard M. Karp and Robert Kleinberg. Noisy binary search and its applications. In SODA, pages 881-890, 2007. URL:
  20. Eduardo Sany Laber and Loana Tito Nogueira. Fast searching in trees. In Eletronic Notes on Discrete Mathematics, 2001. Google Scholar
  21. Shay Mozes, Krzysztof Onak, and Oren Weimann. Finding an optimal tree searching strategy in linear time. In SODA, pages 1096-1105, 2008. URL:
  22. Krzysztof Onak and Pawel Parys. Generalization of binary search: Searching in trees and forest-like partial orders. In FOCS, pages 379-388, 2006. URL:
  23. Andrzej Pelc. Searching games with errors - fifty years of coping with liars. Theor. Comput. Sci., 270(1-2):71-109, 2002. Google Scholar
  24. Alain-Sol Snitzman. Topics in random walks in random environment. ICTP Lecture Notes Series, 2004. Google Scholar
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