Local Routing on Ordered $\Theta$-graphs

TL;DR

This paper proves the non-existence of memoryless local routing algorithms for ordered -graphs and introduces a novel deterministic -memory routing algorithm that guarantees delivery in linear hops.

Contribution

It establishes the impossibility of memoryless routing on ordered -graphs and presents the first deterministic -memory routing algorithm with guaranteed delivery.

Findings

No deterministic memoryless local routing algorithm exists for ordered -graphs.

A new deterministic -memory routing algorithm guarantees delivery from source to destination.

The routing algorithm converges in O(n) hops, where n is the number of vertices.

Abstract

The problem of locally routing on geometric networks using limited memory is extensively studied in computational geometry. We consider one particular graph, the ordered $\Theta$-graph, which is significantly harder to route on than the $\Theta$-graph, for which a number of routing algorithms are known. Currently, no local routing algorithm is known for the ordered $\Theta$-graph. We prove that, unfortunately, there does not exist a deterministic memoryless local routing algorithm that works on the ordered $\Theta$-graph. This motivates us to consider allowing a small amount of memory, and we present a deterministic $O(1)$-memory local routing algorithm that successfully routes from the source to the destination on the ordered $\Theta$-graph. We show that our local routing algorithm converges to the destination in $O(n)$ hops, where $n$ is the number of vertices. To the best of our knowledge, our algorithm is the first deterministic local routing algorithm that is guaranteed to reach the destination on the ordered $\Theta$-graph.