Drawing maps with advice

We study the problem of the amount of information required to draw a complete or a partial map of a graph with unlabeled nodes and arbitrarily labeled ports. A mobile agent, starting at any node of an unknown connected graph and walking in it, has to accomplish one of the following tasks: draw a complete map of the graph, i.e., find an isomorphic copy of it including port numbering, or draw a partial map, i.e., a spanning tree, again with port numbering. The agent executes a deterministic algorithm and cannot mark visited nodes in any way. None of these map drawing tasks is feasible without any additional information, unless the graph is a tree. Hence we investigate the minimum number of bits of information (minimum size of advice  ) that has to be given to the agent to complete these tasks. It turns out that this minimum size of advice depends on the number nn of nodes or the number mm of edges of the graph, and on a crucial parameter μμ, called the multiplicity of the graph, which measures the number of nodes that have an identical view of the graph.We give bounds on the minimum size of advice for both above tasks. For μ=1μ=1 our bounds are asymptotically tight for both tasks and show that the minimum size of advice is very small. For μ>1μ>1 the minimum size of advice increases abruptly. In this case our bounds are asymptotically tight for topology recognition and asymptotically almost tight for spanning tree construction.

► The tasks are not feasible without additional information, unless the graph is a tree.
► ω(1)ω(1) bits of advice are enough for both tasks and Θ(1)Θ(1) bits are not enough if μ=1μ=1.
► Θ(mlogμ)Θ(mlogμ) bits of advice are enough and necessary to recognize topology.
► Ω(μlog(n/μ))Ω(μlog(n/μ)) bits of advice are necessary to construct a spanning tree.
► O(μlogn)O(μlogn) bits of advice are enough to construct a spanning tree.