On a family of strong geometric spanners that admit local routing strategies

Abstract

We introduce a family of directed geometric graphs, whose vertices are points in R d . The graphs G λ θ in this family depend on two real parameters λ and θ. For 1 2 < λ < 1 and π 3 < θ < π 2 , the graph G λ θ is a strong t-spanner for t = 1 ( 1 − λ ) cos θ . That is, for any two vertices p and q, G λ θ contains a path from p to q of length at most t times the Euclidean distance | p q | , and all edges on this path have length at most | p q | . The out-degree of any node in the graph G λ θ is O ( 1 / ϕ d − 1 ) , where ϕ = min ( θ , arccos 1 2 λ ) . We show that routing on G λ θ can be achieved locally. Finally, we show that all strong t-spanners are also t-spanners of the unit-disk graph.