Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees

• Authors: Hsueh-I Lu
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2010

We address the problem of designing compact routing tables for an unlabeled connected $n$-node planar network $G$. For each node $r$ of $G$, the designer is given a routing spanning tree $T_r$ of $G$ rooted at $r$, which specifies the routes for sending packets from $r$ to the rest of $G$. Each node $r$ of $G$ is equipped with ports $1,2,\ldots,\mathit{deg}_r$, where $\mathit{deg}_r$ is the degree of $r$ in $T_r$. Each port of $r$ is supposed to be assigned to a neighbor of $r$ in $T_r$ in a one-to-one manner. For each node $v$ of $G$ with $v\neq r$, let $\mathit{port}_r(v)$ be the port to which $r$ should forward packets with destination $v$. Under the assumption that the designer has the freedom to determine the label and the port assignment of each node in $G$, the routing table design problem is to design a compact routing table $R_r$ for each node $r$ such that $\mathit{port}_r(v)$ can be determined merely from $R_r$ and the label of $v$. Compact routing tables for various network topologies have been extensively studied in the literature. Planar networks are particularly important for routing with geometric metrics. Based upon four-page decompositions of $G$, Gavoille and Hanusse gave the best previously known polynomial-time computable result for this problem with linear-space routing tables, where the time complexity is measured under the conventional unit-cost RAM model of computation: Each $\mathit{port}_r(v)$ is computable from $R_r$ and the label of $v$ in $O(\log^{2+\epsilon}n)$ time for any positive constant $\epsilon$. The number of bits required to encode each $R_r$ is at most $8n+o(n)$. The time required to compute each $R_r$ is $O(n)$. Based on orderly spanning trees of $G$, our design achieves the following improved bounds without increasing the time complexity for computing each $R_r$: Each $\mathit{port}_r(v)$ is computable from $R_r$ and the label of $v$ in $O(\log^{1+\epsilon}n)$ time for any positive constant $\epsilon$. The number of bits required to encode each $R_r$ is at most $7.181n+o(n)$. The overall code length of all $n$ routing tables is at most $7n^2+o(n^2)$ bits.