Voronoi-based splinegon decomposition and shortest-path tree computation

Abstract

In motion planning, two-dimensional (2D) splinegons are typically used to represent the contours of 2D objects. In general, a 2D splinegon must be pre-decomposed to support rapid queries of the shortest paths or visibility. Herein, we propose a new region decomposition strategy, known as the Voronoi-based decomposition (VBD), based on the Voronoi diagram of curved boundary-segment generators (either convex or concave). The number of regions in the VBD is O(n+$c_1$). Compared with the well-established horizontal visibility decomposition (HVD), whose complexity is O(n+$c_2$), the VBD decomposition generally contains less regions because $c_1$≤$c_2$, where n is the number of the vertices of the input splinegon, and $c_1$ and $c_2$ are the number of inserted vertices at the boundary. We systematically discuss the usage of VBD. Based on the VBD, the shortest path tree (SPT) can be computed in linear time. Statistics show that the VBD performs faster than HVD in SPT computations. Additionally, based on the SPT, we design algorithms that can rapidly compute the visibility between two points, the visible area of a point/line-segment, and the shortest path between two points.