Interval routing in some planar networks

In this article, we design optimal or near optimal interval routing schemes (IRS, for short) with small compactness for several classes of plane quadrangulations and triangulations (by optimality or near optimality we mean that messages are routed via shortest or almost shortest paths). We show that the subgraphs of the rectilinear grid bounded by simple circuits allow optimal IRS with at most two circular intervals per edge (2-IRS). We extend this result to all plane quadrangulations in which all inner vertices have degrees4. Namely, we establish that every such graph has an optimal IRS with at most seven linear intervals per edge (7-LIRS). This leads to a 7-LIRS with the stretch factor 2 for all plane triangulations in which all inner vertices have degrees6. All routing schemes can be implemented in linear time.