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 𝑆𝑃_𝑤(s,t) from s to t in the continuous 2-dimensional space, a shortest vertex path 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 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 ‖𝑆𝐺𝑃_𝑤(s,t)‖/‖𝑆𝑃_𝑤(s,t)‖, ‖𝑆𝑉𝑃_𝑤(s,t)‖/‖𝑆𝑃_𝑤(s,t)‖, ‖𝑆𝐺𝑃_𝑤(s,t)‖/‖𝑆𝑉𝑃_𝑤(s,t)‖ 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 ‖𝑆𝐺𝑃_𝑤(s,t)‖/‖𝑆𝑃_𝑤(s,t)‖ is at most 2/√(2+√(2))≈ 1.08 and 2/√(2+√(3))≈ 1.04 in a square and a hexagonal mesh, respectively.