Approximate distance oracles for graphs with dense clusters

Let TeX be a collection of N pairwise vertex disjoint TeX -spanners where the weight of an edge is equal to the Euclidean distance between its endpoints. Let TeX be a graph on TeX with M edges of non-negative weight. The union of the two graphs is denoted TeX . We present a data structure of size TeX that answers (1+e)-approximate shortest path queries in TeX in constant time, where e>¿0 is constant