Approximate distance oracles for graphs with dense clusters

Abstract

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