eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-12-06
48:1
48:13
10.4230/LIPIcs.ISAAC.2018.48
article
Deterministic Treasure Hunt in the Plane with Angular Hints
Bouchard, Sébastien
1
Dieudonné, Yoann
2
Pelc, Andrzej
3
Petit, Franck
1
Sorbonne Université, CNRS, INRIA, LIP6, F-75005 Paris, France
Laboratoire MIS, Université de Picardie Jules Verne, Amiens, France
Département d'informatique, Université du Québec en Outaouais, Gatineau, Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol123-isaac2018/LIPIcs.ISAAC.2018.48/LIPIcs.ISAAC.2018.48.pdf
treasure hunt
deterministic algorithm
mobile agent
hint
plane