Local Routing on Ordered $Theta$-graphs

2025-06-19

AI Summary

Deterministic memoryless local routing on ordered Θ-graphs has long lacked a viable solution—no deterministic memoryless algorithm existed, nor any finite-memory algorithm with theoretical convergence guarantees. Method: We first prove the nonexistence of deterministic memoryless local routing algorithms on ordered Θ-graphs. Then, we design the first deterministic, constant-memory (O(1)-state) local routing algorithm, leveraging angular partitioning, adaptive neighbor selection, and a carefully constructed finite-state machine. Results: Our algorithm guarantees delivery to the target in O(n) hops and is provably correct under rigorous theoretical analysis. It is the first locally routable scheme for ordered Θ-graphs with formal correctness guarantees, thereby resolving a fundamental open problem in geometric graph routing theory and filling a critical gap in the understanding of local routing on ordered Θ-graphs.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Local Search
• Intelligent Robots: Localization, Mapping, and Navigation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

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.

Problem

• Local routing on ordered Θ-graphs with limited memory
• No existing deterministic memoryless routing for ordered Θ-graphs
• Proposing first deterministic O(1)-memory routing for ordered Θ-graphs

Innovation

• Deterministic O(1)-memory local routing algorithm
• Works on ordered Θ-graphs
• Converges in O(n) hops