Approximate distance oracles for graphs with dense clusters

Let H1=(V,E1) be a collection of N pairwise vertex disjoint O(1)-spanners where the weight of an edge is equal to the Euclidean distance between its endpoints. Let H2=(V,E2) be the graph on V with M edges of non-negative weight. The union of the two graphs is denoted G=(V,E1∪E2). We present a data structure of size O(M2+nlogn) that answers (1+ε)-approximate shortest path queries in G in constant time, where ε>0 is constant.