The Higher-Order Voronoi Diagram of Line Segments

Abstract

The order- $$k$$ Voronoi diagram of line segments has properties surprisingly different from its counterpart for points. For example, a single order- $$k$$ Voronoi region may consist of $$\varOmega (n)$$ disjoint faces. In this paper, we analyze the structural properties of this diagram and show that its combinatorial complexity is $$O(k(n-k))$$ , for $$n$$ non-crossing line segments, despite the presence of disconnected regions. The same bound holds for $$n$$ intersecting line segments, for $$k\ge n/2$$ . We also consider the order- $$k$$ Voronoi diagram of line segments that form a planar straight-line graph, and augment the definition of an order- $$k$$ diagram to cover sites that are not disjoint. On the algorithmic side, we extend the iterative approach to construct this diagram, handling complications caused by the presence of disconnected regions. All bounds are valid in the general $$L_p$$ metric, $$1\le p\le \infty $$ . For non-crossing segments in the $$L_\infty $$ and $$L_1$$ metrics, we show a tighter $$O((n-k)^2)$$ bound for $$k>n/2$$ .