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."