Voronoi Diagrams

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Applications of Voronoi Diagrams

Now that we understand the structure of Voronoi diagrams, we're ready to go over the contexts that the IB exam will ask you to apply them in.

Nearest Neighbor Interpolation

Weather stations measure the temperature (in degrees Fahrenheit) throughout the desert based on temperatures taken in five base camps. It's not realistic to set up more stations throughout the desert due to its vast size and remote location. When scientists are studying the behavior of plants and animals in the desert, they use the measurement closest to them as an estimate for the actual temperature of a more remote location.

For example, a scientist located anywhere in the bottom right region would record the temperature as 91°, even if that is not the exact temperature at their specific location. Similarly, a scientist anywhere in the middle region would record the temperature as 82°. A scientist in between two or more regions (on an edge or vertex) would record the temperature as an average of the regions around them, adding the values of the temperatures in the nearest cells and dividing by the number of cells bordering them (two for an edge, three or more for a vertex).

This method is a simple kind of interpolation, a process by which known data points are used to estimate the value of unknown data points.

Nearest neighbor interpolation

Nearest neighbor interpolation is a method for estimating unknown data points based on a Voronoi diagram. If each site is assigned a numerical value, such as the precise temperature at the location of a site, then all other points in that site's cell are also approximated at that value. Points which are equidistant from two are more sites are approximated by averaging the values of the sites it is equidistant from.

Example (temperature):

The graphic shows a heat map overlaid with a Voronoi diagram split into five sections. The generating sites, denoted with black triangles, represent the exact temperatures taken at those locations. Dragging the thermostat around indicates the nearest neighbor approximation (the text on the bottom left) for the location at the thermostat's current position.

Checkpoint

Weather stations measure the temperature (in degrees Fahrenheit) throughout the desert based on temperatures taken in five base camps. The temperatures in other locations are approximated using nearest neighbor interpolation.

State the approximate temperature at site P.

Select the correct option

Toxic Waste Dump Problem

The other application that the IB often asks about is known as the toxic waste dump problem. The goal of this problem is to find the point on the Voronoi diagram that is farthest from all other sites. It's called the toxic waste dump problem because one common context is finding where, in a given area, waste should be dumped so that it is as far as possible from all towns.

Toxic waste dump problems

The toxic waste dump problem, sometimes referred to as the largest empty circle problem, is a type of question that asks you to identify the farthest location from all other given sites. This is solved by constructing a Voronoi diagram, and the answer to the toxic waste dump problem is always a vertex of the corresponding Voronoi diagram. If there are multiple vertices, the solution is the vertex which is further from its closest sites (the one which is the center of a circle with the largest radius).

In the example above, Q is further from its closest points than P, so the solution to this toxic waste dump problem is the point Q.

This kind of problem may also appear in the context of placing businesses as far as possible from their competitors, or identifying the point at the center of the largest circle that can be drawn on a diagram without containing any of the sites.

Discussion

Why is the solution to toxic waste dump problems always located at a vertex of a Voronoi diagram?
How do you determine which vertex the solution should be located at?

Part (a)

We want to choose a point P so that its distance to the nearest toxic site is as large as possible. In other words, we look for P that maximises “how far its closest site is.”

Consider three possibilities for where P might lie in the Voronoi diagram:

P in the interior of a single cell (say the cell of site A). Then A is its unique closest site, and by moving P slightly away from A (still inside that cell) we increase its distance to A—and hence increase its “distance to the nearest site.” So an interior point cannot be optimal.
P on an edge (the perpendicular bisector of A and B). Here P is equally close to A and B, and further from all others. But we can slide P perpendicularly “outwards” from the segment AB so that its distance to both A and B increases—again improving the distance to the closest site. So no edge‐point is a global maximum.
P at a vertex of the Voronoi diagram, where three (or more) sites A, B, C all lie at the same distance from P. Any small move in any direction makes P closer to at least one of A, B or C, thus decreasing its minimum distance. Hence a vertex is a local—and in fact global—maximum for the minimum distance.

Therefore the best spot for the toxic‐waste dump must lie at a Voronoi vertex, where it is “pinned” equally by three or more sites and no slight move can increase its distance to the nearest site.

Part (b)

For each Voronoi vertex you have its coordinates (xV,yV) and you know which three sites meet there — call one of them A with coordinates (xA,yA). Because at a vertex the distances to those three sites are all the same, you only need to compute the distance to one of them. Then repeat for every vertex and pick the one whose distance is the greatest.

Step 1 – Compute the distance at one vertex

dV=√(xV−xA)2+(yV−yA)2

Here (xV,yV) is the vertex and (xA,yA) is any one of its three equidistant sites.

Step 2 – Do the same for the other vertices

If the next vertex has coordinates (xW,yW) and meets site B at (xB,yB), compute

dW=√(xW−xB)2+(yW−yB)2

and so on for every vertex.

Step 3 – Compare your distances

Look at all your computed distances dV,dW,⋯. The vertex whose distance is the largest is the one that is farthest from its nearest toxic site. That vertex is where you should place the dump.

Exercise

The following graphic shows the locations of five coffee shops in a town, labeled A−D and plotted on a Voronoi diagram. The points labeled P1 and P2 are the vertices of the diagram.

Graham wants to open a new coffee shop in town in a location as far as possible from all others.

State the coordinates of the location where Graham should open his coffee shop.

Select the correct option

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