Approximate distance oracles for graphs with dense clusters

Let H-1 = (V, F-1) 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 H-2 = (V, F-2) be a graph on V with M edges of non-negative weight. The union of the two graphs is denoted G = (V, F-1 boolean OR F-2). We present a data structure of size O(M-2 + V log V) that answers (1 + epsilon)-approximate shortest path queries in G in constant time, where epsilon > 0 is constant