The emergence of sparse spanners and well-separated pair decomposition under anarchy

A spanner graph on a set of points in R^<em>d</em> provides shortest paths between any pair of points with lengths at most a constant factor of their Euclidean distance. A spanner with a sparse set of edges is thus a good candidate for network backbones, as desired in many practical scenarios such as the transportation network and peer-to-peer network overlays. In this paper we investigate new models and aim to interpret why good spanners ‘emerge’ in reality, when they are clearly built in pieces by agents with their own interests and the construction is not coordinated. We show that the following algorithm of constructing an edge <em>pq</em>, if and only if there is no existing edge <em>p</em>′<em>q</em>′ with <em>p</em>′ and <em>q</em>′ at distances no more than (4(1+1/ε))^-1·|<em>pq</em>| from <em>p</em>, <em>q</em>respectively, generates a (1+ε)-spanner with a linear number of edges. The algorithm also implies a simple algorithm for constructing linear-size well-separated pair decompositions that may be of interest on their own. This new spanner construction algorithm has applications in the construction of nice network topologies for peer-to-peer systems, when peers join and leave the network and have only limited information about the rest of the network