Introduction

Voronoi diagrams are a fundamental construct in computational geometry and are used extensively in various fields such as computer graphics, geography, and optimization. The intuitive concept behind a Voronoi diagram is the partitioning of a plane into regions based on distance to a specified set of sites (points). Each region corresponds to one of the sites and consists of all points closer to that site than to any other. The Voronoi diagram is thus a graphical representation of such a division.

When it comes to implementing Voronoi diagrams, various algorithms exist. Among the simplest and easiest to implement is Fortune's algorithm, which efficiently constructs a Voronoi diagram in $O(n \log n)$ time.

Overview of Fortune's Algorithm

Fortune's algorithm, developed by Steven Fortune in 1986, is a sweep line algorithm for generating Voronoi diagrams. The fundamental idea is to sweep a line across the plane and incrementally construct the Voronoi diagram by managing a beach line—a curve consisting of parabolic arcs. Here's a step-by-step explanation of the algorithm:

1. Preparation: • Initialize an event queue with all the site points, which are the initial events. • Create an empty status structure (often implemented as a balanced tree) to represent the beach line and a dictionary to handle circle events.
2. Processing Events: • Site Event: When the sweep line encounters a site, add the corresponding arc to the beach line. This may involve breaking an existing arc and creating new breakpoints (vertices of the Voronoi diagram). • Circle Event: When the circle event occurs, it signals that an arc will disappear from the beach line, creating a vertex and connecting edges in the Voronoi diagram.
3. Edge Maintenance: • Keep track of the edges being formed as the algorithm progresses. When a circle event occurs, finalize the edges connected at the removed arc.
4. Completion: • Once all events in the priority queue are processed, finish the edges that go to infinity and handle edge cases on the boundary of the sweep line.

Key Characteristics

Fortune's algorithm is notable for its efficiency and elegance. It reduces the need for complicated geometric calculations into manageable sweep line operations.

Data Structures

• Event Queue: Typically implemented as a priority queue (a min-heap) to efficiently process events ordered by the y-coordinate. • Beach Line: A dynamic data structure (such as a balanced binary tree) that maintains the order of the parabolas intersecting the sweep line. • Voronoi Edges: A structure, often a list or set, to store the edges of the Voronoi diagram.

Algorithm Efficiency

Example

Consider a set of points in a 2D plane: $P = {(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)}$. Using Fortune’s algorithm, the steps can be visualized as follows:

1. Initialize the event queue with all points in $P$.
2. Start the sweep line from the top (positive infinity on the y-axis).
3. For each site event, modify the beach line, adjust the event queue for any new circle events, and add edges to the Voronoi diagram.
4. For circle events, update the beach line, finalize the edges between disappearing arcs, and record any Voronoi vertices created.
5. Continue until the event queue is empty.

Here’s a pseudo-code representation: