The End 2016 Mathematics A To Z: Voronoi Diagram
This is one I never heard of before grad school. And not my first year in grad school either; I was pretty well past the point I should’ve been out of grad school before I remember hearing of it, somehow. I can’t explain that.
Take a sheet of paper. Draw two dots on it. Anywhere you like. It’s your paper. But here’s the obvious thing: you can divide the paper into the parts of it that are nearer to the first, or that are nearer to the second. Yes, yes, I see you saying there’s also a line that’s exactly the same distance between the two and shouldn’t that be a third part? Fine, go ahead. We’ll be drawing that in anyway. But here we’ve got a piece of paper and two dots and this line dividing it into two chunks.
Now drop in a third point. Now every point on your paper might be closer to the first, or closer to the second, or closer to the third. Or, yeah, it might be on an edge equidistant between two of those points. Maybe even equidistant to all three points. It’s not guaranteed there is such a “triple point”, but if you weren’t picking points to cause trouble there probably is. You get the page divided up into three regions that you say are coming together in a triangle before realizing that no, it’s a Y intersection. Or else the regions are three strips and they don’t come together at all.
What if you have four points … You should get four regions. They might all come together in one grand intersection. Or they might come together at weird angles, two and three regions touching each other. You might get a weird one where there’s a triangle in the center and three regions that go off to the edge of the paper. Or all sorts of fun little abstract flag icons, maybe. It’s hard to say. If we had, say, 26 points all sorts of weird things could happen.
These weird things are Voronoi Diagrams. They’re a partition of some surface. Usually it’s a plane or some well-behaved subset of the plane like a sheet of paper. The partitioning is into polygons. Exactly one of the points you start with is inside each of the polygons. And everything else inside that polygon is nearer to its one contained starting point than it is any other point. All you need for the diagram are your original points and the edges dividing spots between them. But the thing begs to be colored. Give in to it and you have your own, abstract, stained-glass window pattern. So I’m glad to give you some useful mathematics to play with.
Voronoi diagrams turn up naturally whenever you want to divide up space by the shortest route to get something. Sometimes this is literally so. For example, a radio picking up two FM signals will switch to the stronger of the two. That’s what the superheterodyne does. If the two signals are transmitted with equal strength, then the receiver will pick up on whichever the nearer signal is. And unless the other mathematicians who’ve talked about this were just as misinformed, cell phones pick which signal tower to communicate with by which one has the stronger signal. If you could look at what tower your cell phone communicates with as you move around, you would produce a Voronoi diagram of cell phone towers in your area.
Mathematicians hoping to get credit for a good thing may also bring up Dr John Snow’s famous halting of an 1854 cholera epidemic in London. He did this by tracking cholera outbreaks and measuring their proximity to public water pumps. He shut down the water pump at the center of the severest outbreak and the epidemic soon stopped. One could claim this as a triumph for Voronoi diagrams, although Snow can not have had this tool in mind. Georgy Voronoy (yes, the spelling isn’t consistent. Fashions in transliterating Eastern European names — Voronoy was Ukrainian and worked in Warsaw when Poland was part of the Russian Empire — have changed over the years) wasn’t even born until 1868. And it doesn’t require great mathematical insight to look for the things an infected population has in common. But mathematicians need some tales of heroism too. And it isn’t as though we’ve run out of epidemics with sources that need tracking down.
Voronoi diagrams turned out to be useful in my own meager research. I needed to model the flow of a fluid over a whole planet, but could only do so with a modest number of points to represent the whole thing. Scattering points over the planet was easy enough. To represent the fluid over the whole planet as a collection of single values at a couple hundred points required this Voronoi-diagram type division. … Well, it used them anyway. I suppose there might have been other ways. But I’d just learned about them and was happy to find a reason to use them. Anyway, this is the sort of technique often used to turn information about a single point into approximate information about a region.
(And I discover some amusing connections here. Voronoy’s thesis advisor was Andrey Markov, who’s the person being named by “Markov Chains”. You know those as those predictive-word things that are kind of amusing for a while. Markov Chains were part of the tool I used to scatter points over the whole planet. Also, Voronoy’s thesis was On A Generalization Of A Continuous Fraction, so, hi, Gaurish! … And one of Voronoy’s doctoral students was Wacław Sierpiński, famous for fractals and normal numbers.)
Voronoi diagrams have a lot of beauty to them. Some of it is subtle. Take a point inside its polygon and look to a neighboring polygon. Where is the representative point inside that neighbor polygon? … There’s only one place it can be. It’s got to be exactly as far as the original point is from the edge between them, and it’s got to be on the direction perpendicular to the edge between them. It’s where you’d see the reflection of the original point if the border between them were a mirror. And that has to apply to all the polygons and their neighbors.
From there it’s a short step to wondering: imagine you knew the edges. The mirrors. But you don’t know the original points. Could you figure out where the representative points must be to fit that diagram? … Or at least some points where they may be? This is the inverse problem, and it’s how I first encountered them. This inverse problem allows nice stuff like algorithm compression. Remember my description of the result of a Voronoi diagram being a stained glass window image? There’s no reason a stained glass image can’t be quite good, if we have enough points and enough gradations of color. And storing a bunch of points and the color for the region is probably less demanding than storing the color information for every point in the original image.
If we want images. Many kinds of data turn out to work pretty much like pictures, set up right.