Efficient Construction of Low Complexity Geodesic Spanners in Polygonal Domains

Core Concepts

The authors present efficient algorithms to construct low complexity geodesic spanners for a set of n point sites in a polygonal domain with m vertices. They introduce the novel spanner property of complexity, which is the total complexity of all edges in the spanner, and provide trade-offs between the spanning ratio and the spanner complexity.

Abstract

The authors study the problem of constructing geometric spanners for a set of n point sites in a constrained two-dimensional environment, such as a polygonal domain. A geometric t-spanner is an edge-weighted graph where the (weighted) distance between any two sites is at most t times their original distance.

The key contributions are:

1. For a simple polygon P with m vertices:

   • They present an algorithm to construct a 2√2k-spanner with complexity O(mn^(1/k) + n log^2 n) in O(n log^2 n + m log n + K) time, where K is the output complexity.
   • They show how to refine this spanner to obtain a (t + ε)-spanner with the same complexity.

2. For a polygonal domain P with h holes:

   • They construct a relaxed geodesic 6k-spanner of complexity O(mn^(1/k) + n log^2 n) in O((n + m) log^2 n log m + K) time.

3. They provide lower bounds on the complexity of (relaxed) geodesic spanners, showing that the spanning ratio of their spanners is near-optimal.

The authors also present efficient algorithms to compute the required components, such as balanced shortest-path separators in polygonal domains, which may be of independent interest.

Stats

For a simple polygon P with m vertices and a set S of n point sites:

Any (3 - ε)-spanner may have complexity Ω(nm).
For any constant ε ∈ (0, 1) and integer constant t ≥ 2, there exists a set of n sites in a simple polygon P with m = Ω(n) vertices for which any (relaxed) geodesic (t - ε)-spanner has complexity Ω(mn^(1/(t-1))).

Quotes

"For any constant ε ∈ (0, 1) and integer constant t ≥ 2, there exists a set of n point sites in a simple polygon P with m = Ω(n) vertices for which any (relaxed) geodesic (t - ε)-spanner has complexity Ω(mn^(1/(t-1)))."

Key Insights Distilled From

The Complexity of Geodesic Spanners

by Sarita de Be... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2303.02997.pdf

Deeper Inquiries

What are some real-world applications where low complexity geodesic spanners could be particularly useful

Low complexity geodesic spanners have various real-world applications where efficient network design is crucial. One such application is in transportation networks, where the construction of railway or road networks needs to balance the trade-off between the number of connections and the distances between locations. By using low complexity geodesic spanners, transportation networks can be designed with fewer connections while ensuring that the distances between locations are within acceptable detour limits. This can lead to more efficient and cost-effective transportation systems.
Another application is in communication networks, where the design of communication links between nodes in a network is essential. Low complexity geodesic spanners can help in reducing the complexity of the network while maintaining the connectivity and communication efficiency between nodes. This is particularly useful in scenarios where the network needs to be scalable and easily maintainable.
Furthermore, in robotics and autonomous systems, geodesic spanners with low complexity can be utilized for path planning and obstacle avoidance. By constructing efficient paths between locations with minimal complexity, robots and autonomous vehicles can navigate through complex environments more effectively and safely.

How could the techniques developed in this paper be extended to handle dynamic updates to the set of point sites or the polygonal domain

To handle dynamic updates in the set of point sites or the polygonal domain, the techniques developed in the paper can be extended by incorporating dynamic data structures and algorithms. One approach could be to implement data structures like dynamic trees or segment trees to efficiently update the spanner structure when new points are added or removed from the set of sites. Additionally, techniques from computational geometry, such as incremental algorithms for geometric problems, can be employed to update the spanner in response to changes in the polygonal domain.
By maintaining data structures that support dynamic updates, the spanner can be continuously adjusted to reflect changes in the environment or the set of point sites. This would enable the spanner to adapt in real-time to modifications in the network topology or the spatial configuration of the domain, ensuring that the spanner remains accurate and efficient even with dynamic updates.

Are there any other interesting properties of geodesic spanners, beyond spanning ratio and complexity, that could be worth exploring in future research

Beyond spanning ratio and complexity, there are several other interesting properties of geodesic spanners that could be worth exploring in future research. Some of these properties include:

Fault-tolerance: Investigating the resilience of geodesic spanners to node or edge failures, and developing algorithms to maintain connectivity and efficiency in the presence of faults.
Robustness to changes in the environment: Studying how geodesic spanners can adapt to changes in the underlying geometry or topology of the domain, and developing methods to ensure robustness in dynamic environments.
Scalability: Exploring the scalability of geodesic spanners to large-scale networks or complex domains, and developing scalable algorithms for constructing and maintaining spanners in such scenarios.
Energy efficiency: Analyzing the energy consumption of geodesic spanners in communication networks or sensor networks, and optimizing the spanner structure to minimize energy usage while maintaining connectivity and efficiency.

By investigating these properties, researchers can gain a deeper understanding of the behavior and characteristics of geodesic spanners, leading to the development of more efficient and adaptable network structures for various applications.