On approximating shortest paths in weighted triangular tessellations

Abstract

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. [Path-length analysis for grid-based path planning. Artificial Intelligence, 301:103560, 2021], we prove upper and lower bounds on the ratios $\frac{\Vert SG{P}_w(s,t)\Vert }{\Vert S{P}_w(s,t)\Vert }$, $\frac{\Vert SV{P}_w(s,t)\Vert }{\Vert S{P}_w(s,t)\Vert }$, $\frac{\Vert SG{P}_w(s,t)\Vert }{\Vert SV{P}_w(s,t)\Vert }$, 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 $\frac{\Vert SG{P}_w(s,t)\Vert }{\Vert S{P}_w(s,t)\Vert }=\frac{2}{\sqrt{3}}\approx 1.15$ 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 $\frac{\Vert SV{P}_w(s,t)\Vert }{\Vert S{P}_w(s,t)\Vert }⪅1.15$.

25 pages, 15 figures, accepted version