Asymptotically Optimal Solutions for Small World Graphs

  • Authors:

  • Michele Flammini;Luca Moscardelli;Alfredo Navarra;Stéphane Pérennes

  • Affiliations:

  • University of L’Aquila, Computer Science Department, Via Vetoio, 67100, L’Aquila, Italy;University of L’Aquila, Computer Science Department, Via Vetoio, 67100, L’Aquila, Italy;University of L’Aquila, Computer Science Department, Via Vetoio, 67100, L’Aquila, Italy;I3S-CNRS/INRIA/University of Nice, MASCOTTE project, Via Vetoio, 67100, Sophia Antipolis, France

  • Venue:
  • Theory of Computing Systems
  • Year:
  • 2008

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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 2n) 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.