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

We introduce a family of directed geometric graphs, whose vertices are points in ^Rd. The graphs Gλθ in this family depend on two real parameters λ and θ. For 12<λ<1 and π3<θ<π2, the graph Gλθ is a strong t-spanner for t=1(1-λ)cosθ. That is, for any two vertices p and q, Gλθ contains a path from p to q of length at most t times the Euclidean distance |pq|, and all edges on this path have length at most |pq|. The out-degree of any node in the graph Gλθ is O(1/πd- ¹), where π=min(θ,arccos12λ). We show that routing on Gλθ can be achieved locally. Finally, we show that all strong t-spanners are also t-spanners of the unit-disk graph.