The Emergence of Sparse Spanners and Greedy Well-Separated Pair Decomposition

A spanner graph on a set of points in R contains a shortest path between any pair of points with length 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. Our main result is to show that the following algorithm generates a (1 + ε)-spanner with a linear number of edges, constant average degree, and the total edge length as a small logarithmic factor of the cost of the minimum spanning tree. In our algorithm, the points build edges at an arbitrary order. When a point p checks on whether the edge to a point q should be built, it will build this edge only if there is no existing edge pq with p and q at distances no more than 1 4(1+1/ε) · |p q| from p, q respectively. Eventually when all points have finished checking edges to all other points, the resulted collection of edges forms a sparse spanner as desired. This new spanner construction algorithm can be extended to a metric space with constant doubling dimension and admits a local routing scheme to find the short paths. As a side product, we show a greedy algorithm for constructing linear-size well-separated pair decompositions that may be of interest on its own. A well-separated pair decomposition is a collection of subset pairs such that each pair of point sets is fairly far away from each other compared with their diameters and that every pair of points is ‘covered’ by at least one well-separated pair. Our greedy algorithm selects an arbitrary pair of points that have not yet been covered and puts a ‘dumb-bell’ around the pair as the well-separated pair, repeats this until all pairs of points are covered. When the algorithm finishes, we show only a linear number of pairs is generated, which is asymptotically optimal.