Approximating shortest paths in weighted square and hexagonal meshes

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.