Computing an $L_1$ shortest path among splinegonal obstacles in the plane

Abstract

We reduce the problem of computing an $L_1$ shortest path between two given points s and t in the given splinegonal domain $\mathcal {S}$ to the problem of computing an $L_1$ shortest path between two points in the polygonal domain. Our reduction algorithm defines a polygonal domain $\mathcal {P}$ from $\mathcal {S}$ by identifying a coreset of points on the boundaries of splinegons in $\mathcal {S}$. Further, it transforms a shortest path between s and t among polygonal obstacles in $\mathcal {P}$ to a shortest path between s and t among splinegonal obstacles in $\mathcal {S}$. When $\mathcal {S}$ is comprised of h pairwise disjoint simple splinegons defined with a total of n vertices, excluding the time to compute an $L_1$ shortest path among simple polygonal obstacles in $\mathcal {P}$, our reduction algorithm takes $O(n + h \lg {n} + (\lg {h})^{1+\epsilon })$ time. Here, $\epsilon$ is a small positive constant [resulting from the triangulation of the free space using Bar-Yehuda and Chazelle (Int J Comput Geom Appl 4(4):475–481, 1994)]. For the special case of $\mathcal {S}$ comprising of concave-out splinegons, we have devised another reduction algorithm. This algorithm does not rely on the structures used in the algorithm (Inkulu and Kapoor in Comput Geom 42(9):873–884, 2009) to compute an $L_1$ shortest path in the polygonal domain. Further, we have characterized few of the properties of $L_1$ shortest paths among splinegons which could be of independent interest.