Approximate distance oracles for graphs with dense clusters

Abstract

Let ℋ1 = (Ν, ℱ1) 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 = (Ν, ℱ2) be a graph on Ν with M edges of non-negative weight. The union of the two graphs is denoted = (Ν, ℱ1∪ℱ2). We present a data structure of size O(M2 + |Ν| log |Ν|) that answers (1 + ε)-approximate shortest path queries in in constant time, where ε > 0 is constant. © Springer-Verlag 2004.