Spanners in randomly weighted graphs: Euclidean case.

Abstract Given a connected graph and a length function we let denote the shortest distance between vertex and vertex . A ‐spanner is a subset such that if denotes shortest distances in the subgraph then for all . We study the size of spanners in the following scenario: we consider a random embedding of into the unit square with Euclidean edge lengths. For constant, we prove the existence w.h.p. of ‐spanners for that have edges. These spanners can be constructed in time. (We will use to indicate that the hidden constant depends on ). There are constraints on preventing it going to zero too quickly.