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

TL;DR

This paper explores how sparse spanners and well-separated pair decompositions naturally emerge through simple, local construction rules, demonstrating their efficiency and optimality in geometric graphs.

Contribution

It introduces a new model showing that local, greedy edge addition rules produce near-optimal sparse spanners and well-separated pair decompositions.

Findings

Edges built under the local rule form a (1+ε)-spanner with linear edges

The resulting graph has constant average degree

Total edge length is logarithmically close to the MST cost

Abstract

A spanner graph on a set of points in $R^d$ 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.