On approximating shortest paths in weighted triangular tessellations

TL;DR

This work analyzes how discretizing a weighted plane with an equilateral triangular tessellation affects shortest-path computations. By introducing the crossing path $X(s,t)$ and decomposing interactions into six weakly simple polygons, the authors derive a weight-agnostic upper bound on the ratio $\frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert}$, showing it never exceeds $\frac{2}{\sqrt{3}}$ (approximately 1.15) and proving this bound is tight. A corollary extends the bound to the weighted any-angle path $\mathit{SVP_w}(s,t)$, giving $\frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} \lessapprox \frac{2}{\sqrt{3}}$, while a related analysis yields explicit bounds for ratios involving grid- and vertex-based paths in $G_{6\text{corner}}$ and $G_{\text{corner}}$. The results justify using compact triangular grids for path planning in games and GIS, even under arbitrary nonnegative face weights, and extend prior binary-weight analyses to the general weighted-region setting, with extensions to square and hexagonal tessellations discussed as future work.

Abstract

We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path $ \mathit{SP_w}(s,t) $, which is a shortest path from $ s $ to $ t $ in the space; a weighted shortest vertex path $ \mathit{SVP_w}(s,t) $, which is an any-angle shortest path; and a weighted shortest grid path $ \mathit{SGP_w}(s,t) $, which is a shortest path whose edges are edges of the tessellation. Given any arbitrary weight assignment to the faces of a triangular tessellation, thus extending recent results by Bailey et al. [Path-length analysis for grid-based path planning. Artificial Intelligence, 301:103560, 2021], we prove 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} $, which provide estimates on the quality of the approximation. It turns out, surprisingly, that our worst-case bounds are independent of any weight assignment. Our main result is that $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} = \frac{2}{\sqrt{3}} \approx 1.15 $ in the worst case, and this is tight. As a corollary, for the weighted any-angle path $ \mathit{SVP_w}(s,t) $ we obtain the approximation result $ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} \lessapprox 1.15 $.

Figures (15)

• Figure 1: Screenshots of the "Colossal Citadels" game by Uneven Dungeon. Used with permission from the author. Note the triangular grid underlying the scene.
• Figure 2: Vertex $v$ is connected to its neighbors in a triangular tessellation. The dashed lines represent the edges of the graphs that coincide with the edges of the cells.
• Figure 3: $\mathit{SP_w}(s,t)$ (blue), $\mathit{SVP_w}(s,t)$ (green), and a $\mathit{SGP_w}(s,t)$ (red) between two corners $s$ and $t$ in $G_{6\text{corner}}$. The cost of each path is $16.75$, $17.32$ and $18$, respectively, for a cell side length of $2$.
• Figure 4: When the centers of the cells are used as the vertices of the associated graph, we can make the ratios $\frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert}, \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert}$ arbitrarily large by giving cells $T_4, T_5, T_6$ a finite weight much greater than 1, and cells $T_1, T_3$ weight $1$.
• Figure 5: Weighted shortest path $\mathit{SP_w}(s,t)$ (blue) and the crossing path $X(s,t)$ (orange) from $s$ to $t$ in a triangular tessellation.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 3
• Definition 4
• Lemma 3
• proof