Spanners for Geometric Intersection Graphs

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R, a (1+ǫ)-spanner with O(nǫ) edges is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.