Cost vs. Information Tradeoffs for Treasure Hunt in the Plane

Abstract: A mobile agent has to find an inert treasure hidden in the plane. Both the agent and the treasure are modeled as points. This is a variant of the task known as treasure hunt. The treasure is at a distance at most $D$ from the initial position of the agent, and the agent finds the treasure when it gets at distance $r$ from it, called the {\em vision radius}. However, the agent does not know the location of the treasure and does not know the parameters $D$ and $r$. The cost of finding the treasure is the length of the trajectory of the agent. We investigate the tradeoffs between the amount of information held {\em a priori} by the agent and the cost of treasure hunt. Following the well-established paradigm of {\em algorithms with advice}, this information is given to the agent in advance as a binary string, by an oracle cooperating with the agent and knowing the location of the treasure and the initial position of the agent. The size of advice given to the agent is the length of this binary string. For any size $z$ of advice and any $D$ and $r$, let $OPT(z,D,r)$ be the optimal cost of finding the treasure for parameters $z$, $D$ and $r$, if the agent has only an advice string of length $z$ as input. We design treasure hunt algorithms working with advice of size $z$ at cost $O(OPT(z,D,r))$ whenever $r\leq 1$ or $r\geq 0.9D$. For intermediate values of $r$, i.e., $1<r\<0.9D$, we design an almost optimal scheme of algorithms: for any constant $\alpha\>0$, the treasure can be found at cost $O(OPT(z,D,r)^{1+\alpha})$.