From 0c3cb6a19d1307f317a94a9569c1c50705de229f Mon Sep 17 00:00:00 2001 From: Clement Jamin Date: Wed, 4 Jan 2017 16:35:39 +0100 Subject: [PATCH] Update and clarify complexity --- .../doc/Triangulation/Triangulation.txt | 20 ++++++++++--------- 1 file changed, 11 insertions(+), 9 deletions(-) diff --git a/Triangulation/doc/Triangulation/Triangulation.txt b/Triangulation/doc/Triangulation/Triangulation.txt index b4ac0504e0e..12c63333fd2 100644 --- a/Triangulation/doc/Triangulation/Triangulation.txt +++ b/Triangulation/doc/Triangulation/Triangulation.txt @@ -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