On approximating shortest paths in weighted triangular tessellations
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: aweighted shortest path SPw(s, t),...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universidad de Alcalá (UAH) |
| Repositorio: | e_Buah Biblioteca Digital Universidad de Alcalá |
| Idioma: | inglés |
| OAI Identifier: | oai:ebuah.uah.es:10017/64219 |
| Acceso en línea: | http://hdl.handle.net/10017/64219 https://dx.doi.org/10.1016/j.artint.2023.103898 |
| Access Level: | acceso abierto |
| Palabra clave: | Shortest path Any-angle path planning Tessellation Weighted region problem Matemáticas Mathematics |
| Sumario: | 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: aweighted shortest path SPw(s, t), which is a shortest path from s to t in the space; aweighted shortest vertex path SVPw(s, t), which is an any-angle shortest path; and a weighted shortest grid path SGPw(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 SGPw(s,t) SPw(s,t) , SVPw(s,t) SPw(s,t) , SGPw(s,t) SVPw(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 SGPw(s,t) √ SPw(s,t) = 2 3 ≈1.15 in the worst case, and this is tight. As a corollary, for the weighted any-angle path SVPw(s, t) we obtain the approximation result SVPw(s,t) SPw(s,t) ⪅ 1.15. |
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