Abstract
We consider the problem of determining constructions with an asymptotically optimal oblivious diameter in small world graphs under the Kleinberg’s model. In particular, we give the first general lower bound holding for any monotone distance distribution, that is induced by a monotone generating function. Namely, we prove that the expected oblivious diameter is Ω(log 2 n) even on a path of n nodes. We then focus on deterministic constructions and after showing that the problem of minimizing the oblivious diameter is generally intractable, we give asymptotically optimal solutions, that is with a logarithmic oblivious diameter, for paths, trees and Cartesian products of graphs, including d-dimensional grids for any fixed value of d.
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The research was partially funded by the European project COST Action 293, “Graphs and Algorithms in Communication Networks” (GRAAL).
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Flammini, M., Moscardelli, L., Navarra, A. et al. Asymptotically Optimal Solutions for Small World Graphs. Theory Comput Syst 42, 632–650 (2008). https://doi.org/10.1007/s00224-007-9073-y
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DOI: https://doi.org/10.1007/s00224-007-9073-y