The Higher-Order Voronoi Diagram of Line Segments

Surprisingly, the order-$k$ Voronoi diagram of line segments had received no attention in the computational-geometry literature. It illustrates properties surprisingly different from its counterpart for points; for example, a single order-$k$ Voronoi region may consist of $\Omega(n)$ disjoint faces. We analyze the structural properties of this diagram and show that its combinatorial complexity for $n$ non-crossing line segments is $O(k(n-k))$, despite the disconnected regions. The same bound holds for $n$ intersecting line segments, when $k\geq n/2$. 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-$k$ Voronoi diagram to cover non-disjoint sites, addressing the issue of non-uniqueness for $k$-nearest sites. Furthermore, we enhance the iterative approach to construct this diagram. All bounds are valid in the general $L_p$ metric, $1\leq p\leq \infty$. For non-crossing segments in the $L_\infty$ and $L_1$ metrics, we show a tighter $O((n-k)^2)$ bound for $k>n/2$.