Piecemeal Learning of an Unknown Environment
Machine Learning - Special issue on COLT '93
Computing on Anonymous Networks: Part I-Characterizing the Solvable Cases
IEEE Transactions on Parallel and Distributed Systems
Exploring Unknown Environments
SIAM Journal on Computing
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Compact labeling schemes for ancestor queries
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Labeling schemes for flow and connectivity
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
The power of a pebble: exploring and mapping directed graphs
Information and Computation
An Effective Characterization of Computability in Anonymous Networks
DISC '01 Proceedings of the 15th International Conference on Distributed Computing
Local and global properties in networks of processors (Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Tree exploration with little memory
Journal of Algorithms
Journal of the ACM (JACM)
Optimal graph exploration without good maps
Theoretical Computer Science
Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing
Oracle size: a new measure of difficulty for communication tasks
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Local MST computation with short advice
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Tree exploration with logarithmic memory
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Trade-offs between the size of advice and broadcasting time in trees
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
Journal of Graph Theory
Fast Radio Broadcasting with Advice
SIROCCO '08 Proceedings of the 15th international colloquium on Structural Information and Communication Complexity
Deterministic Rendezvous in Trees with Little Memory
DISC '08 Proceedings of the 22nd international symposium on Distributed Computing
Information and Computation
SIROCCO'07 Proceedings of the 14th international conference on Structural information and communication complexity
Label-guided graph exploration by a finite automaton
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Toward more localized local algorithms: removing assumptions concerning global knowledge
Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing
Perspective: Simple agents learn to find their way: An introduction on mapping polygons
Discrete Applied Mathematics
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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. This is due to the impossibility of recognizing already visited nodes. 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 numbers n of nodes or the number m of edges of the graph, and on a crucial parameter µ, 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 = 1 our bounds are asymptotically tight for both tasks and show that the minimum size of advice is very small: for an arbitrary function ϕ = ω(1) it suffices to give ϕ(n) bits of advice to accomplish both tasks for n-node graphs, and Θ(1) bits are not enough. For µ 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. We show that Θ(m log µ) bits of advice are enough and necessary to recognize topology in the class of graphs with m edges and multiplicity µ 1. For the second task we show that Ω(µ log(n/µ)) bits of advice are necessary and Ω(µ log n) bits of advice are enough to construct a spanning tree in the class of graphs with n nodes and multiplicity µ 1. Thus in this case the gap between our bounds is always at most logarithmic, and the bounds are asymptotically tight for multiplicity µ = O(nα), where α is any constant smaller than 1.