Edge Exploration of a Graph by Mobile Agent

Abstract

In this paper, we study the problem of edge exploration of an n node graph by a mobile agent. The nodes of the graph are unlabeled, and the ports at a node of degree d are arbitrarily numbered $0,\dots , d-1$. A mobile agent, starting from some node, has to visit all the edges of the graph and stop. The time of the exploration is the number of edges the agent traverses before it stops. The task of exploration can not be performed even for a class of cycles if no additional information, called advice, is provided to the agent a priori. Following the paradigm of algorithms with advice, this priori information is provided to the agent by an Oracle in the form of a binary string. The Oracle knows the graph, but does not have the knowledge of the starting point of the agent. In this paper, we consider the following two problems of edge exploration. The first problem is: “how fast is it possible to explore an n node graph regardless of the size of advice provided to the agent?”

We show a lower bound of $\Omega (n^{\frac{8}{3}})$ on exploration time to answer the above question. Next, we show the existence of an $O(n^3)$ time algorithm with $O(n \log n)$ advice. The second problem then asks the following question: “what is the smallest advice that needs to be provided to the agent in order to achieve time $O(n^3)$?” We show a lower bound $\Omega (n^\delta )$ on size of the advice, for any $\delta <\frac{1}{3}$, to answer the above question.