Spanners for Geometric Intersection Graphs

TL;DR

This paper introduces efficient algorithms for constructing geometric spanners in intersection graphs, leveraging space partitioning and bichromatic closest pair solutions, with applications to proximity problems.

Contribution

It presents novel algorithms for building spanners in geometric intersection graphs with near-optimal complexity and extends these methods to arbitrary ball graphs.

Findings

Constructed (1+ε)-spanners for unit ball graphs in R^k.

Achieved sub-quadratic running time for arbitrary ball graphs.

Provided efficient algorithms for proximity problems like diameter and distance queries.

Abstract

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner 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.