# Computing a Rectilinear Shortest Path amid Splinegons in Plane

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

## Abstract

We reduce the problem of computing a rectilinear shortest path between two given points s and t in the given splinegonal domain $$\mathcal {S}$$ to the problem of computing a rectilinear 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 amid polygonal obstacles in $$\mathcal {P}$$ to a shortest path between s and t amid splinegonal obstacles in $$\mathcal {S}$$. When $$\mathcal {S}$$ is comprised of h pairwise disjoint splinegons defined with a total of n vertices, excluding the time to compute a rectilinear shortest path amid polygons 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 ). For the special case of $$\mathcal {S}$$ comprising concave-in splinegons, we have devised another reduction algorithm which does not rely on the structures used in the algorithm (Inkulu and Kapoor ) to compute a rectilinear shortest path in the polygonal domain. Further, we have characterized few of the properties of rectilinear shortest paths amid splinegons which could be of independent interest.

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