Compact Routing on Chordal Rings of Degree 4

Abstract.

A chordal ring G(n;c) of degree 4 is a ring of n nodes with chords connecting each vertex i to the vertex (i + c) mod n . In this paper we investigate compact routing schemes on such networks. We show an optimal boolean routing scheme for any such network that requires O( log n) bits of storage at each node, and O(1) time to compute a shortest path to any destination. This improves on the results of [16] which gives a linear time algorithm for such networks and [6] where efficient routing schemes for certain fixed values of c were developed.

Further, we show several bounds on interval routing schemes for such networks. We show that while every chordal ring has an optimal interval routing scheme with at most $2\sqrt{n}$ intervals on any edge, there exist chordal rings for which any optimal interval routing scheme that labels the vertices around the ring in the graph requires $\Omega (\sqrt{n})$ intervals on some edges. Additionally, there are chordal rings which admit no optimal one-interval routing schemes, regardless of the vertex labeling. We also consider interval routing schemes under relaxed requirements for the lengths of paths.