Approximating shortest paths in weighted square and hexagonal meshes

TL;DR

The paper investigates how weighted square and hexagonal meshes approximate continuous 2D shortest-path distances by comparing the true continuous optimum SP_w(s,t) with grid-constrained paths: SGP_w(s,t) and SVP_w(s,t). Using crossing paths X(s,t) and carefully defined polygon types, it derives weight-independent upper bounds on the critical ratio R = \lVert SGP_w(s,t) \rVert / \lVert SP_w(s,t) \rVert for two main mesh families: square meshes with 8-corner connectivity yield R ≤ 2/\sqrt{2+\sqrt{2}} ≈ 1.08, while hexagonal meshes with 12-corner connectivity yield R ≤ 2/\sqrt{2+\sqrt{3}} ≈ 1.04. The work also binds related ratios involving SVP_w and SP_w across square and hex graphs, including a 1.04 upper bound for SVP_w/ SP_w in hex meshes and a 1.50 bound in certain hex configurations, with a complementary 1.03 lower bound in the corner-graph setting. Overall, the results provide tight, weight-agnostic guarantees for grid-based shortest-path approximations across all regular tessellations, with implications for algorithm design and potential 3D extensions.

Abstract

Continuous 2-dimensional space is often discretized by considering a mesh of weighted cells. In this work we study how well a weighted mesh approximates the space, with respect to shortest paths. We consider a shortest path $ \mathit{SP_w}(s,t) $ from $ s $ to $ t $ in the continuous 2-dimensional space, a shortest vertex path $ \mathit{SVP_w}(s,t) $ (or any-angle path), which is a shortest path where the vertices of the path are vertices of the mesh, and a shortest grid path $ \mathit{SGP_w}(s,t) $, which is a shortest path in a graph associated to the weighted mesh. We provide upper and lower bounds on the ratios $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $, $ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $, $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SVP_w}(s,t)\rVert} $ in square and hexagonal meshes, extending previous results for triangular grids. These ratios determine the effectiveness of existing algorithms that compute shortest paths on the graphs obtained from the grids. Our main results are that the ratio $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $ is at most $ \frac{2}{\sqrt{2+\sqrt{2}}} \approx 1.08 $ and $ \frac{2}{\sqrt{2+\sqrt{3}}} \approx 1.04 $ in a square and a hexagonal mesh, respectively.

Figures (28)

• Figure 1: Note the regular meshes underlying the scenes.
• Figure 2: Vertex $v$ is connected to its neighbors in $G_{4\text{corner}}$ (top left), in $G_{8\text{corner}}$ (top right), in $G_{3\text{corner}}$ (bottom left), and in $G_{12\text{corner}}$ (bottom right). The edges of each graph that coincide with the cell edges are depicted with dashed lines.
• Figure 3: Vertex $v$ is connected to some of its neighbors in $G_{\text{corner}}$ of a square (left) and a hexagonal (right) mesh. The edges of each graph that coincide with the cell edges are depicted with dashed lines.
• Figure 4: The weighted length of $\mathit{SP_w}(s,t)$ (blue), $\mathit{SVP_w}(s,t)$ (green), and $\mathit{SGP_w}(s,t)$ (red) is, respectively, $11.26$, $11.41$, and $12$, when the length of the cell side is $1$ in $G_{4\text{corner}}$.
• Figure 5: $\mathit{SGP_w}(s,t)$ (in red) intersects a cell that $\mathit{SP_w}(s,t) = \mathit{SVP_w}(s,t)$ (in blue) does not intersect.

Theorems & Definitions (47)

• Definition 1
• Lemma 2
• proof
• Definition 4
• Proposition 5
• proof
• Definition 6
• Proposition 7
• proof
• Definition 11