Deterministic Treasure Hunt in the Plane with Angular Hints

Abstract

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 $\varTheta (D^2)$. We show that if all angles given as hints are at most $\pi$, then the cost can be lowered to O(D), which is the optimal complexity. 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 complexity $\varTheta (D^2)$ cannot be beaten.