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arXiv · 2506.16021

Local Routing on Ordered $\Theta$-graphs

André van Renssen·Shuei Sakaguchi
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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.

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André van Renssen, Shuei Sakaguchi. 2025-06-19. Local Routing on Ordered $\Theta$-graphs. https://arxiv.org/abs/2506.16021

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