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Fortune's Voronoi algorithm implemented in C#

, 21 Apr 2013 MPL
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A C# implementation of the Fortune algorithm to compute 2D Voronoi graphs.


NOTE: Code has moved to Google Code! 


Given a set of two dimensional vectors (or data points), a Voronoi graph is a separation of those points into compartments where all points inside one compartment are closer to the contained data point than to any other data point. I won't give any demonstration here, but if you want to know more about Voronoi graphs, check out this.

The applications of Voronoi graphs are quite broad. Very useful for a lot of optimization problems (in most cases, the Delaunay Triangulation which can be easily derived from a Vononoi graph is used there), it ranges to computing topological maps from bitmaps.

[This is an article for freaks. After a rather painful experience writing the thing I hope it will benefit everyone who is looking for this algorithm in a civilized language (or simply does not want to use Fortune's original C implementation).]

In 1987, Steve Fortune described an algorithm to compute such a graph by using a sweep line in combination with a binary tree. A PowerPoint explanation of the algorithm (the one I used to implement it) can be found here. Note that I did not use the linked data structure to represent a graph - I think that is an unnecessary difficulty in the age of ArrayLists and HashSets.

The Implementation

Data points are represented by my own Vector class. It can do much more than needed here (but there was no reason to strip it before bringing it) but I won't explain it here. The most important fact is that although working with doubles the Vector class automatically rounds values to 10 digits (or whatever is set in the Vector.Precision field). Yes, sadly, this is very important if you want to sort of compare doubles.

A VoronoiGraph is a class that only contains a HashSet of vertices (as 2D vectors) and a HashSet of VoronoiEdges - each with references to the left and right data point and (of course) the two vertices that bound the edge. If the edge is (partially or completely) unbounded, the vector Fortune.VVUnknown is used.

BinaryPriorityQueue is used for the sweep line event queue.


The algorithm itself (Fortune.ComputeVoronoiGraph(IEnumerable)) takes any IEnumerable containing only two dimensional vectors. It will return a VoronoiGraph. The algorithm's complexity is O(n ld(n)) with a factor of about 10 microseconds on my machine (2GHz).


This article, along with any associated source code and files, is licensed under The Mozilla Public License 1.1 (MPL 1.1)


About the Author

Software Developer (Senior)
Germany Germany
I did my diploma in Dresden and Sydney where I dealt with algorithms, agents and other cool AI stuff. Now I moved to Frankfurt to work on my PhD dealing with software structures for artificial intelligence systems. If I can, I do things in C# and ASP.NET, but if I have to, my C++, Java and SQL are not that bad.
Long Live .NET.

Comments and Discussions

GeneralMy vote of 5 PinprofessionalKristian Lindberg Vinther9-Jan-15 10:42 
QuestionInteresting PinmemberMember 1083802822-May-14 12:58 
AnswerRe: Interesting PinmemberKristian Vinther5-Jan-15 5:32 
QuestionHow to get a Polygon list asociated to seed Points Pinmembermarceloarguello70012-Jan-14 10:17 
QuestionSmall example code would be very much appreciated [modified] Pinmemberpan05417-Dec-13 5:20 
Questionvoronoy bad link PinmemberGiovanni Cuccureddu21-Sep-13 22:22 
Questiondivide and conquer . Pinmemberrizikandry23-Jul-13 4:02 
QuestionMESSAGE FROM THE OWNER: Help improve this project on Google Code! PinmemberBenDi21-Apr-13 7:59 
QuestionRounding error when creating circles for new nodes PinmemberBwd Yeti19-Apr-13 1:50 
On line 350 of FortuneVoronoi.cs in CircleCheckDataNode(), after creating a new circle for three points the code checks if the bottom of the circle is below the most recently added point, since if the circle is above that it would defy the sweep line
However you round the circle y value to 10 decimal places, so in extreme situations where the new node is at the exact bottom of the circle it can have higher precision on its y value and invalidate the circle, throwing off the whole graph. The if statement needs changed to something like
if(VC.Y>ys || Math.Abs(VC.Y - ys) < 1e-10)
to account for it
An example graph exhibiting this problem, with the offending point marked red, before and after fixing
AnswerRe: Rounding error when creating circles for new nodes PinmemberBenDi21-Apr-13 7:56 

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