Abstract:
We study the problem of treasure hunt in a graph by a mobile agent. The nodes in the
graph are anonymous and the edges at any node v of degree deg(v) are labeled arbitrarily
as 0, 1,...,deg(v) − 1. A mobile agent, starting from a node, must find a stationary object,
called treasure that is located on an unknown node at a distance D from its initial position.
The agent finds the treasure when it reaches the node where the treasure is present. The
time of treasure hunt is defined as the number of edges the agent visits before it finds the
treasure. The agent does not have any prior knowledge about the graph or the position
of the treasure. An Oracle, that knows the graph, the initial position of the agent, and the
position of the treasure, places some pebbles on the nodes, at most one per node, of the
graph to guide the agent towards the treasure.
We target to answer the question: what is the fastest possible treasure hunt algorithm
regardless of the number of pebbles are placed?
We show an algorithm that uses O(D log) pebbles to find the treasure in a graph G in
time O(D log), where is the maximum degree of a node in G and D is the distance
from the initial position of the agent to the treasure. We show a matching lower bound of
(D log ) on time of the treasure hunt using any number of pebbles