Local Routing Algorithms on Euclidean Spanners with Small Diameter

Abstract

Given a set of n points in the plane, we present two constructions of geometric r-spanners with $r \ge 1$ based on a hierarchical decomposition. These graphs have ${{\,\textrm{O}\,}}(n)$ edges and diameter ${{\,\textrm{O}\,}}(\log n)$. We then design online routing algorithms on these graphs.

The first construction is based on $\varTheta _k$-graphs (with $k > 6$ and $k\equiv 2 \mod 4$). The routing algorithm is memoryless and local (i.e. it uses information about the closed neighborhood of the current vertex and the destination). It has routing ratio $1/(1-2\sin (\pi /k))$ and finds a path with ${{\,\textrm{O}\,}}(\log ^2 n)$ edges.

The second construction uses a TD-Delaunay triangulation, which is a Delaunay triangulation where the empty regions are homothets of an equilateral triangle. The associated routing algorithm is local and memoryless, has a routing ratio of $5/{\sqrt{3}}$, finds a path consisting of ${{\,\textrm{O}\,}}(\log ^2 n)$ edges and requires the pre-computation of vertex labels of ${{\,\textrm{O}\,}}(\log ^2 n)$ bits (assuming the nodes are placed on a grid of polynomial size).

We have examples that show when using either of our routing algorithms, in the worst case, the paths returned by the algorithm can consist of $\varOmega (\log ^2 n)$ edges.

Research of P. Bose and Y. Garito supported in part by NSERC.