AIJ Journal 2023 Journal Article
On approximating shortest paths in weighted triangular tessellations
- Prosenjit Bose
- Guillermo Esteban
- David Orden
- Rodrigo I. Silveira
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 S P w ( s, t ), which is a shortest path from s to t in the space; a weighted shortest vertex path SV P w ( s, t ), which is an any-angle shortest path; and a weighted shortest grid path SG P 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. (2021) [6], we prove upper and lower bounds on the ratios ‖ SG P w ( s, t ) ‖ ‖ S P w ( s, t ) ‖, ‖ SV P w ( s, t ) ‖ ‖ S P w ( s, t ) ‖, ‖ SG P w ( s, t ) ‖ ‖ SV P w ( s, t ) ‖, 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 ‖ SG P w ( s, t ) ‖ ‖ S P w ( s, t ) ‖ = 2 3 ≈ 1. 15 in the worst case, and this is tight. As a corollary, for the weighted any-angle path SV P w ( s, t ) we obtain the approximation result ‖ SV P w ( s, t ) ‖ ‖ S P w ( s, t ) ‖ ⪅ 1. 15.