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Posted 10 Aug 2005

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# Fortune's Voronoi algorithm implemented in C#

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A C# implementation of the Fortune algorithm to compute 2D Voronoi graphs.

NOTE: Code has moved to Google Code!

## Introduction

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 `ArrayList`s and `HashSet`s.

## 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 `double`s 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 `double`s.

A `VoronoiGraph` is a class that only contains a `HashSet` of vertices (as 2D vectors) and a `HashSet` of `VoronoiEdge`s - 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.

## Usage

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).

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 Software Developer (Senior) 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.

 FirstPrev Next
 A mistake Dimasina4-May-06 11:20 Dimasina 4-May-06 11:20
 Re: A mistake mharley5-May-06 5:58 mharley 5-May-06 5:58
 Re: A mistake Dimasina5-May-06 11:35 Dimasina 5-May-06 11:35
 Re: A mistake BenDi6-May-06 11:17 BenDi 6-May-06 11:17
 Fortune's Voronoi algorithm implemented in C# pc_kf13-Mar-06 10:43 pc_kf 13-Mar-06 10:43
 Re: Fortune's Voronoi algorithm implemented in C# BenDi13-Mar-06 21:29 BenDi 13-Mar-06 21:29
 Voronoi Cell's attributes Truong Pham Dang Khoa10-Mar-06 23:36 Truong Pham Dang Khoa 10-Mar-06 23:36
 Re: Voronoi Cell's attributes BenDi12-Mar-06 11:12 BenDi 12-Mar-06 11:12
 Mmmh, dont have a trivial solution for that, especcially since the cells might be infinite. It's not hard to find out which voronoi vertices are around one data point, you'll just have to look though all the edges and take the vertices of all edges that have the desired point either as left or right data point. From that set of vertices you should be able to compute cell data using some matrix operations, dont have an implementation for that, though.
 Wrong Diagram - What's the problem ? ramirob1-Feb-06 12:19 ramirob 1-Feb-06 12:19
 Re: Wrong Diagram - What's the problem ? BenDi1-Feb-06 22:09 BenDi 1-Feb-06 22:09
 Re: Wrong Diagram - What's the problem ? TomekS19-Mar-06 8:02 TomekS 19-Mar-06 8:02
 Re: Wrong Diagram - What's the problem ? BenDi20-Mar-06 10:16 BenDi 20-Mar-06 10:16
 Semi-unbounded edges darrellp13-Dec-05 9:24 darrellp 13-Dec-05 9:24
 Re: Semi-unbounded edges BenDi13-Dec-05 15:19 BenDi 13-Dec-05 15:19
 Re: Semi-unbounded edges darrellp13-Dec-05 18:40 darrellp 13-Dec-05 18:40
 Re: Semi-unbounded edges Dimasina4-May-06 5:26 Dimasina 4-May-06 5:26