Deterministic Treasure Hunt in the Plane with Angular Hints
A mobile agent equipped with a compass and a measure of length has to find an inert treasure in the Euclidean plane. Both the agent and the treasure are modeled as points. In the beginning, the agent is at a distance at most D>0 from the treasure, but knows neither the distance nor any bound on it. Finding the treasure means getting at distance at most 1 from it. The agent makes a series of moves. Each of them consists in moving straight in a chosen direction at a chosen distance. In the beginning and after each move the agent gets a hint consisting of a positive angle smaller than 2 pi whose vertex is at the current position of the agent and within which the treasure is contained. We investigate the problem of how these hints permit the agent to lower the cost of finding the treasure, using a deterministic algorithm, where the cost is the worst-case total length of the agent's trajectory. It is well known that without any hint the optimal (worst case) cost is Theta(D^2). We show that if all angles given as hints are at most pi, then the cost can be lowered to O(D), which is optimal. If all angles are at most beta, where beta<2 pi is a constant unknown to the agent, then the cost is at most O(D^{2-epsilon}), for some epsilon>0. For both these positive results we present deterministic algorithms achieving the above costs. Finally, if angles given as hints can be arbitrary, smaller than 2 pi, then we show that cost Theta(D^2) cannot be beaten.
treasure hunt
deterministic algorithm
mobile agent
hint
plane
Theory of computation~Design and analysis of algorithms
Computing methodologies~Mobile agents
48:1-48:13
Regular Paper
Sébastien
Bouchard
Sébastien Bouchard
Sorbonne Université, CNRS, INRIA, LIP6, F-75005 Paris, France
Yoann
Dieudonné
Yoann Dieudonné
Laboratoire MIS, Université de Picardie Jules Verne, Amiens, France
Andrzej
Pelc
Andrzej Pelc
Département d'informatique, Université du Québec en Outaouais, Gatineau, Canada
This work was supported in part by NSERC discovery grant 8136 - 2013 and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais.
Franck
Petit
Franck Petit
Sorbonne Université, CNRS, INRIA, LIP6, F-75005 Paris, France
This work was performed within Project ESTATE (Ref. ANR-16-CE25-0009-03), supported by French state funds managed by the ANR (Agence Nationale de la Recherche).
10.4230/LIPIcs.ISAAC.2018.48
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Sébastien Bouchard, Yoann Dieudonné, Andrzej Pelc, and Franck Petit
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