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Almost-Optimal Deterministic Treasure Hunt in Arbitrary Graphs

Authors Sébastien Bouchard, Yoann Dieudonné, Arnaud Labourel , Andrzej Pelc

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Author Details

Sébastien Bouchard
  • Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France
Yoann Dieudonné
  • MIS Lab., Université de Picardie Jules Verne, Amiens, France
Arnaud Labourel
  • Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France
Andrzej Pelc
  • Département d'informatique, Université du Québec en Outaouais, Gatineau, Canada

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Sébastien Bouchard, Yoann Dieudonné, Arnaud Labourel, and Andrzej Pelc. Almost-Optimal Deterministic Treasure Hunt in Arbitrary Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 36:1-36:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


A mobile agent navigating along edges of a simple connected graph, either finite or countably infinite, has to find an inert target (treasure) hidden in one of the nodes. This task is known as treasure hunt. The agent has no a priori knowledge of the graph, of the location of the treasure or of the initial distance to it. The cost of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until finding the treasure. Awerbuch, Betke, Rivest and Singh [Baruch Awerbuch et al., 1999] considered graph exploration and treasure hunt for finite graphs in a restricted model where the agent has a fuel tank that can be replenished only at the starting node s. The size of the tank is B = 2(1+α)r, for some positive real constant α, where r, called the radius of the graph, is the maximum distance from s to any other node. The tank of size B allows the agent to make at most {⌊ B⌋} edge traversals between two consecutive visits at node s. Let e(d) be the number of edges whose at least one extremity is at distance less than d from s. Awerbuch, Betke, Rivest and Singh [Baruch Awerbuch et al., 1999] conjectured that it is impossible to find a treasure hidden in a node at distance at most d at cost nearly linear in e(d). We first design a deterministic treasure hunt algorithm working in the model without any restrictions on the moves of the agent at cost 𝒪(e(d) log d), and then show how to modify this algorithm to work in the model from [Baruch Awerbuch et al., 1999] with the same complexity. Thus we refute the above twenty-year-old conjecture. We observe that no treasure hunt algorithm can beat cost Θ(e(d)) for all graphs and thus our algorithms are also almost optimal.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • treasure hunt
  • graph
  • mobile agent


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