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Update and clarify complexity

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Clement Jamin 2017-01-04 16:35:39 +01:00
parent 3704a59422
commit 0c3cb6a19d
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Triangulation/doc/Triangulation/Triangulation.txt
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@ -508,16 +508,18 @@ This simple example shows how to create a regular triangulation.
\section TriangulationSecPerf Complexity and Performances
When inserting a set of points into a Delaunay triangulation,
the current implementation starts
by spatially sorting the points. Then, for each point to insert,
it locates it by walking in the triangulation, using the
previously inserted vertex as a "hint". Finally, the point is
When inserting a batch of points into a Delaunay triangulation,
the current implementation starts by spatially sorting the points.
Then, for each point to insert, it locates it by walking in the triangulation,
using the previously inserted vertex as a "hint". Finally, the point is
inserted.
In the worst case, the expected complexity is
\f$ O(n^{\lceil\frac{d}{2}\rceil}+n\log n)\f$. When the algorithm is
run on \f$ n \f$ random points, the cost of inserting one point is
\f$ O(n^{1/d}) \f$.
In the worst case scenario, without spatial sort, the expected complexity is
\f$ O(n^{\lceil\frac{d}{2}\rceil+1}) \f$.
When the algorithm is run on uniformly distributed points, the localization complexity is
\f$ O(n^{\frac{1}{d}}) \f$ and the size of the triangulation is \f$ O(n) \f$, which gives
a complexity of \f$ O(n^{1+\frac{1}{d}}) \f$ for the insertion.
With spatial sort and random points, one can expect a complexity of \f$ O(n\log n) \f$.
Please refer to \cgalCite{boissonnat2009Delaunay} for more details.
We provide below (\cgalFigureRef{Triangulationfigbenchmarks100},
\cgalFigureRef{Triangulationfigbenchmarks1000} and