On the complexity of greedy routing in ring-based peer-to-peer networks

ABSTRACT

We investigate the complexity of greedy routing in uniform ring-based random graphs, a general model that captures many topologies that have been proposed for peer-to-peer and social networks. In this model the nodes form a ring; for each node u we independently draw the set of distances along the ring from u to its "long-range contacts" from a fixed distribution P (the same for all and connect u to the corresponding nodes as well as its ring successor. We prove that, for any distribution P, in a graph with n nodes and an expected number of long-range contacts per node constructed in this fashion, the expected number of steps for greedy routing is Ω((log²n)/la^log*n), for some constant a > 1. This improves an earlier lower bound of Ω((log²n)/llog log n) by Aspnes et al. and is very close to the upper bound of O((log²n)/l) achieved by greedy routing in Kleinberg's (one-dimensional) "small-world" networks, a particular instance of uniform ring-based random graphs.

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INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Routing and layout

Additional Classification:
  C. Computer Systems Organization
  C.2 COMPUTER-COMMUNICATION NETWORKS
      C.2.2 Network Protocols
          Subjects: Routing protocols

  E. Data
  E.1 DATA STRUCTURES
      Subjects: Graphs and networks

  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory
          Subjects: Graph algorithms; Network problems

General Terms:
Algorithms, Performance, Theory

Keywords:
coupling, greedy routing, lower bound, peer-to-peer, random graphs, small-worlds