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Bounding the Approximation Error of Shortest Paths in Weighted Square and Hexagonal Meshes


Core Concepts
The ratio of the length of the shortest grid path to the length of the actual shortest path in a weighted square or hexagonal mesh is bounded above by a constant factor, independent of the weight assignment.
Abstract
The paper studies the approximation error when computing shortest paths in weighted square and hexagonal meshes, compared to the actual shortest path in the continuous 2D space. Key highlights: The authors define three types of shortest paths: the actual shortest path SPw(s,t), the shortest vertex path SVPw(s,t), and the shortest grid path SGPw(s,t). They provide upper and lower bounds on the ratios ∥SGPw(s,t)∥/∥SPw(s,t)∥, ∥SVPw(s,t)∥/∥SPw(s,t)∥, and ∥SGPw(s,t)∥/∥SVPw(s,t)∥ for square and hexagonal meshes. The main results are that the ratio ∥SGPw(s,t)∥/∥SPw(s,t)∥ is at most 2√2+√2 ≈ 1.08 for square meshes and 2√2+√3 ≈ 1.04 for hexagonal meshes, independent of the weight assignment. The analysis involves defining "crossing paths" and "simple polygons" formed by the intersection of the different paths, and bounding the ratios within these polygons. The authors also discuss how their results generalize and improve upon previous work on triangular meshes.
Stats
∥SGPw(s,t)∥/∥SPw(s,t)∥ ≤ 2√2+√2 ≈ 1.08 for square meshes ∥SGPw(s,t)∥/∥SPw(s,t)∥ ≤ 2√2+√3 ≈ 1.04 for hexagonal meshes
Quotes
"The ratio ∥SGPw(s,t)∥/∥SPw(s,t)∥ is at most 2√2+√2 ≈ 1.08 when any (non-negative) weight is assigned to the cells of a square mesh and each vertex is connected to its 8 neighboring vertices, and R ≤ 2√2+√3 ≈ 1.04 when each vertex is connected to its 12 neighboring vertices in a hexagonal mesh."

Deeper Inquiries

How do the bounds change if the vertices are placed at the centers of the cells instead of the corners

If the vertices are placed at the centers of the cells instead of the corners in a square mesh, the bounds on the ratios of the lengths of the different types of shortest paths may change. When the vertices are at the centers, the paths may follow a different trajectory compared to when the vertices are at the corners. This change in trajectory can affect the lengths of the paths and consequently impact the ratios between them. The analysis would need to be adjusted to account for this change in vertex placement.

Can the analysis be extended to other types of polygonal tessellations beyond squares and hexagons

The analysis presented in the context can potentially be extended to other types of polygonal tessellations beyond squares and hexagons. The key lies in adapting the methodology to the specific characteristics of the tessellations. For example, for triangular tessellations, the approach may need to consider different types of polygons and the intersections between the paths and the tessellation edges. By modifying the definitions and properties of the polygons and paths, the analysis can be extended to different types of polygonal tessellations.

What are the practical implications of these theoretical bounds in terms of path planning algorithms and their performance in real-world applications

The theoretical bounds derived from the analysis of the ratios between different types of shortest paths in weighted square and hexagonal meshes have practical implications for path planning algorithms in real-world applications. These bounds provide insights into the effectiveness of using grid paths or vertex paths as approximations to the actual shortest paths. By understanding the approximation errors and the factors influencing them, designers can make informed decisions when selecting path planning algorithms for applications such as geographic information systems, robotics, or gaming. The bounds help in evaluating the trade-offs between computational efficiency and solution quality, guiding the implementation of efficient path planning algorithms in various scenarios.
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