On approximating shortest paths in weighted triangular tessellations
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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 𝑆𝑃_𝑤(s,t), which is a shortest path from s to t in the space; a weighted shortest vertex path 𝑆𝑉𝑃_𝑤(s,t), which is a shortest path where the vertices of the path are vertices of the tessellation; and a weighted shortest grid path 𝑆𝐺𝑃_𝑤(s,t), which is a shortest path whose edges are edges of the tessellation. 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)‖ provide estimates on the quality of the approximation. Given any arbitrary weight assignment to the faces of a triangular tessellation, we prove upper and lower bounds on the estimates that are independent of the weight assignment. Our main result is that ‖𝑆𝐺𝑃_𝑤(s,t)‖/‖𝑆𝑃_𝑤(s,t)‖ = 2/√(3)≈ 1.15 in the worst case, and this is tight.
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