Lecture 16: Fortune's Algorithm and Voronoi diagrams
Voronoi Diagrams: Voronoi diagrams (like convex hulls) are among the most important structures in computational geometry. A Voronoi diagram records information about what is close to what.

Definition:

Let P = {p₁,p₂, ... , p_n} be a set of points in the plane (or in any dimensional space), which we call sites.
Define V(p_i), the Voronoi cell for p_i, to be the set of points q in the plane that are closer to p_i than to any other site. That is, the Voronoi cell for p_i is defined to be: V(p_i ) = {q | dist(p_i,q) < dist(p_j, q), for j != i}: The Voronoi diagram of a set of points is the planar subdivision whose faces are the voronoi cells of the points.

Because we need to define "distance", we need to now talk about euclidean geometry, this is a subtle but important shift. Up to know, virtually everything that we have done has not needed the notion of angles, lengths, or distances (except for our work on circles). All geometric tests were made on the basis of orientation tests, a purely affine construct. But there are important geometric algorithms which depend on nonaffine quantities such as distances and angles. Let us begin by defining the Euclidean length of a vector v = (v_x, v_y) in the plane to be |v| = sqrt(x² + y²) The distance between two points p and q, denoted dist(p; q), is defined to be |p - q|. (recall that in affine geometry, we weren't allowed to do the "-" operation on two points. Euclidean geometry allows this).

Alternative Definition:

Another way to define V(p_i) is in terms of the intersection of halfplanes. Given two sites p_i and p_j, the set of points that are strictly closer to p_i than to p_j is just the open halfplane whose bounding line is the perpendicular bisector between p_i and p_j. Denote this halfplane h(p_i, p_j). It is easy to see that a point q lies in V(p_i) if and only if q lies within the intersection of h(p_i, p_j) for all j != i. In other words, V(p_i) = Intersection (for all j) h(p_i, p_j): Since the intersection of halfplanes is a (possibly unbounded) convex polygon, it is easy to see that V(p_i) is a (possibly unbounded) convex polygon. Finally, define the Voronoi diagram of P , denoted Vor(P ) to be what is left of the plane after we remove all the (open) Voronoi cells. It is not hard to prove (see the text) that the Voronoi diagram consists of a collection of line segments which may be unbounded, either at one end or both.

Figure 56: Voronoi diagram
Voronoi diagrams have a number of important applications:

Nearest neighbor queries: One of the most important data structures problems in com putational geometry is solving nearest neighbor queries. Given a point set P , and given a query point q, determine the closest point in P to q. This can be answered by first computing a Voronoi diagram and then locating the cell of the diagram that contains q. (We have already discussed point location algorithms.)
Computational morphology: Some of the most important operations in morphology (used very much in computer vision) is that of ``growing'' and ``shrinking'' (or ``thinning'') objects. If we grow a collection of points, by imagining a grass fire starting simultaneously from each point, then the places where the grass fires meet will be along the Voronoi diagram. The medial axis of a shape (used in computer vision) is just a Voronoi diagram of its boundary.
Facility location: We want to open a new Blockbuster video. It should be placed as far as possible from any existing video stores? Where should it be placed? It turns out that the vertices of the Voronoi diagram are the points that locally at maximum distances from any other point in the set.
High clearance path planning: A robot wants to move around a set of obstacles. To mini mize the possibility of collisions, it should stay as far away from the obstacles as possible. To do this, it should walk along the edges of the Voronoi diagram.

Properties of the Voronoi diagram:

Here are some observations about the structure of Voronoi diagrams in the plane.

Voronoi edges: Each point on an edge of the Voronoi diagram is equidistant from its two nearest neighbors p_i and p_j. Thus, there is a circle centered at such a point such that p_i and p_j lie on this circle, and no other site is interior to the circle.
Voronoi vertices: It follows that vertex at which three Voronoi cells V(p_i), V(p_j), and V(p_k) intersect is equidistant from all sites. Thus it is the center of the circle passing through these sites, and this circle contains no other sites in its interior.
Degree: If we assume that no four points are cocircular, then the vertices of the Voronoi diagram all have degree three.
Convex hull: A cell of the Voronoi diagram is unbounded if and only if the corresponding site lies on the convex hull. (Observe that a point is on the convex hull if and only if it is the closest point from some point at infinity.)
Size: If n denotes the number of sites, then the Voronoi diagram is a planar graph (if we imagine all the unbounded edges as going to a common vertex infinity) with exactly n faces. It follows from Euler's formula that the number of Voronoi vertices is at most 2n - 5 and the number of edges is at most 3n - 6. (See the text for details.)

Computing Voronoi Diagrams: There are a number of algorithms for computing Voronoi diagrams. Of course, there is a naive O(n² log n) time algorithm, which operates by computing V(p_i) by intersecting the n - 1 bisector halfplanes h(p_i, p_j), for j != i. However, there are much more efficient ways, which run in O(n log n) time. Since the convex hull can be extracted from the Voronoi diagram in O(n) time, it follows that this is asymptotically optimal in the worst case. Historically, O(n²) algorithms for computing Voronoi diagrams were known for many years (based on incremental constructions). When computational geometry came along, a more complex, but asymptotically superior O(n log n) algorithm was discovered. This algorithm was based on divide and conquer. But it was rather complex, and somewhat difficult to understand. Later, Steven Fortune invented a plane sweep algorithm for the problem, which provided a simpler O(n log n) solution to the problem. It is his algorithm that we will discuss. Somewhat later still, it was discovered that the incremental algorithm is actually quite efficient, if it is run as a randomized incremental algorithm. We will discuss this algorithm later when we talk about the dual structure, called a Delaunay triangulation.

Images / Examples

Real World Voronoi
voronoi game
Voro-glide