Impact of Knowledge on the Cost of Treasure Hunt in Trees

A mobile agent has to find an inert target in some environment that can be a graph or a terrain in the plane. This task is known as treasure hunt. We consider deterministic algorithms for treasure hunt in trees. Our goal is to establish the impact of different kinds of initial knowledge given to the agent on the cost of treasure hunt, defined as the total number of edge traversals until the agent reaches the treasure hidden in some node of the tree. The agent can be initially given either a complete map of the tree rooted at its starting node, with all port numbers marked, or a blind map of the tree rooted at its starting node but without port numbers. It may also be given, or not, the distance from the root to the treasure. This yields four different knowledge types that are partially ordered by their precision. (For example knowing the blind map and the distance is less precise than knowing the complete map and the distance). The penalty of a less precise knowledge type ${\cal T}_2$ over a more precise knowledge type ${\cal T}_1$ measures intuitively the worst-case ratio of the cost of an algorithm supplied with knowledge of type ${\cal T}_2$ over the cost of an algorithm supplied with knowledge of type ${\cal T}_1$. Our main results establish penalties for comparable knowledge types in this partial order. For knowledge types with known distance, the penalty for having a blind map over a complete map turns out to be very large. By contrast, for unknown distance, the penalty of having a blind map over having a complete map is small. When a map is provided (either complete or blind), the penalty of not knowing the distance over knowing it is medium.