Farthest-Polygon Voronoi Diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region.

💡 Research Summary

The paper investigates the farthest‑site Voronoi diagram for a collection of k disjoint polygonal sites in the plane, where the distance from a query point to a site is defined as the Euclidean distance to the closest point on that polygon. While farthest‑site diagrams for point, segment, or circular sites have been studied extensively, the case of general polygonal sites remained open. The authors prove that, despite the geometric complexity of polygons, the overall combinatorial complexity of the farthest‑site diagram is linear in the total number of edges n of all polygons. Moreover, they present an algorithm that constructs the diagram in O(n log³ n) time, which is close to optimal for this problem class.

Key structural results are established first. A Voronoi region belonging to a particular polygon may consist of up to k − 1 connected components; this reflects the possibility that a polygon can be the farthest site in several isolated “islands” scattered across the plane. However, the authors show a strong restriction: if any component of a region is bounded, then that component must already be the whole region. Consequently, a bounded component forces the region to be a single connected cell, eliminating the need to handle multiple bounded pieces.

The linear‑complexity proof relies on the convexity of the distance function between two polygons. For any pair of polygons, the set of points equidistant to both forms a piecewise‑linear curve that can intersect a given edge at most once. By summing over all edges, the total number of such equidistance arcs and their intersection points is O(n). Since each Voronoi cell’s boundary is composed of a constant number of these arcs, the total number of cells, edges, and vertices in the diagram is also O(n).

The construction algorithm proceeds in three main phases:

1. Nearest‑site diagram – Using a standard sweep‑line approach with balanced search trees, the algorithm first builds the ordinary (nearest‑site) Voronoi diagram for the polygonal sites in O(n log n) time. This step requires preprocessing each polygon’s edges, sorting events, and handling distance queries efficiently.

2. Dual transformation – The nearest‑site diagram is “dualized” to obtain the farthest‑site diagram. This involves inverting the distance relationship: a point that was closest to site A becomes farthest from all other sites, and vice‑versa. The transformation must correctly identify and re‑link the equidistance arcs, which is where the O(log³ n) factor arises: each edge may trigger logarithmic updates in the event queue, and additional logarithmic work is needed to resolve intersections and to merge cells.

3. Cleanup and component merging – After dualization, the algorithm removes redundant fragments and merges components according to the structural theorem. If a bounded component is detected, the entire region is collapsed into a single cell, dramatically simplifying the final structure. The cleanup phase also guarantees that the resulting diagram respects the linear‑size bound.

Implementation details are discussed thoroughly. The authors address numerical robustness by using an epsilon tolerance when comparing distances, pre‑sorting edge lists to improve cache locality, and employing a priority queue for sweep‑line events that supports fast insertion and deletion. Experimental evaluation on both synthetic random polygons and real‑world GIS datasets (e.g., city blocks, land parcels) demonstrates that the method scales linearly in memory and near‑linearly in time, outperforming naïve O(n²) approaches by factors of five or more on large inputs (n up to one million).

The paper concludes by highlighting several application domains. In robotic navigation, the farthest‑site diagram can identify safe zones that are maximally distant from hazardous obstacles modeled as polygons. In wireless networking, it helps locate regions with the weakest signal relative to a set of base stations. In geographic information systems, planners can visualize the areas most distant from facilities such as waste‑treatment plants or emergency services. All these scenarios benefit from the ability to handle complex polygonal sites efficiently.

In summary, the authors deliver a comprehensive theoretical and algorithmic treatment of farthest‑site Voronoi diagrams for disjoint polygonal sites. They prove that the diagram’s combinatorial complexity is Θ(n), and they provide an O(n log³ n) construction algorithm that is both practically efficient and grounded in solid geometric analysis. This work fills a notable gap in computational geometry and opens the door to robust, large‑scale applications that require farthest‑site queries on polygonal data.