4 research outputs found
The Emergence of Sparse Spanners and Greedy Well-Separated Pair Decomposition
A spanner graph on a set of points in contains a shortest path between
any pair of points with length at most a constant factor of their Euclidean
distance. 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 if edges are built in an arbitrary order but an edge is
built if and only if its endpoints are not 'close' to the endpoints of an
existing edge, the graph is a (1 + \eps)-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. As a side product,
we show a simple greedy algorithm for constructing optimal size well-separated
pair decompositions that may be of interest on its own