# Routing in Polygonal Domains

## TL;DR

This paper presents a routing scheme in polygonal domains that guarantees paths close to shortest with efficient labels and routing tables, suitable for networks with complex polygonal obstacles.

## Contribution

It introduces a routing scheme for polygonal domains with near-optimal path length, small labels, and routing tables, improving routing efficiency in complex environments.

## Key findings

- Routing paths within a factor of 1 + ε of shortest paths.
- Labels are of size O(log n) bits.
- Routing tables are of size O((ε^{-1}+h) log n).

## Abstract

We consider the problem of routing a data packet through the visibility graph of a polygonal domain $P$ with $n$ vertices and $h$ holes. We may preprocess $P$ to obtain a label and a routing table for each vertex of $P$. Then, we must be able to route a data packet between any two vertices $p$ and $q$ of $P$, where each step must use only the label of the target node $q$ and the routing table of the current node. For any fixed $\varepsilon > 0$, we present a routing scheme that always achieves a routing path whose length exceeds the shortest path by a factor of at most $1 + \varepsilon$. The labels have $O(\log n)$ bits, and the routing tables are of size $O((\varepsilon^{-1}+h)\log n)$. The preprocessing time is $O(n^2\log n)$. It can be improved to $O(n^2)$ for simple polygons.