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.
From: André van Renssen [view email]
[v1]
Thu, 19 Jun 2025 04:39:13 UTC (507 KB)
[v2]
Tue, 11 Aug 2026 02:58:04 UTC (395 KB)
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