Consider the problem of discovering (or verifying) the edges and non-edges of a network, modelled as a connected undirected graph, using a minimum number of queries. A query at a vertex v discovers (or verifies) all edges and non-edges whose endpoints have different distance from v. In the network discovery problem, the edges and non-edges are initially unknown, and the algorithm must select the next query based only on the results of previous queries. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio O(sqrt(n*log n)) for graphs with n vertices. We also show that no deterministic algorithm can have competitive ratio better than 3. In the network verification problem, the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and non-edges. This problem has previously been studied as the problem of placing landmarks in graphs or determining the metric dimension of a graph. We show that there is no approximation algorithm for this problem with ratio o(log n) unless P=NP. Furthermore, we prove that the optimal number of queries for d-dimensional hypercubes is Theta(d/log d).
@InProceedings{beerliova_et_al:DagSemProc.05031.17, author = {Beerliova, Zuzana and Eberhard, Felix and Erlebach, Thomas and Hall, Alexander and Hoffmann, Michael and Mihalak, Matus and Ram, L. Shankar}, title = {{Network Discovery and Verification}}, booktitle = {Algorithms for Optimization with Incomplete Information}, pages = {1--4}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {5031}, editor = {Susanne Albers and Rolf H. M\"{o}hring and Georg Ch. Pflug and R\"{u}diger Schultz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05031.17}, URN = {urn:nbn:de:0030-drops-594}, doi = {10.4230/DagSemProc.05031.17}, annote = {Keywords: on-line algorithms , set cover , landmarks , metric dimension} }
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