From 49cc30ead71d0656778e4ef29bec5e31b23d529c Mon Sep 17 00:00:00 2001 From: Sylvain Pion Date: Sat, 3 Jan 2009 11:38:41 +0000 Subject: [PATCH] Remove useless \ccTexHtml around math mode $$. --- .../Triangulation_2/triangulation_user.tex | 24 +++++++++---------- .../Delaunay_triangulation_2.tex | 14 +++++------ .../Triangulation_2_ref/Triangulation_2.tex | 10 ++++---- 3 files changed, 24 insertions(+), 24 deletions(-) diff --git a/Triangulation_2/doc_tex/Triangulation_2/triangulation_user.tex b/Triangulation_2/doc_tex/Triangulation_2/triangulation_user.tex index e67d9850ff4..12be5a6ee1e 100644 --- a/Triangulation_2/doc_tex/Triangulation_2/triangulation_user.tex +++ b/Triangulation_2/doc_tex/Triangulation_2/triangulation_user.tex @@ -462,7 +462,7 @@ begins at a vertex of the face which is given as an optional argument or at an arbitrary vertex of the triangulation if no optional argument is given. It takes -time \ccTexHtml{$O(n)$}{O(n)} in the worst case, but only \ccTexHtml{$O(\sqrt{n})$}{O(sqrt(n))} +time $O(n)$ in the worst case, but only $O(\sqrt{n})$ on average if the vertices are distributed uniformly at random. The class \ccc{Triangulation_hierarchy_2}, described in section~\ref{Section_2D_Triangulations_Hierarchy}, @@ -473,14 +473,14 @@ Insertion of a point is done by locating a face that contains the point, and splitting this face into three new faces. If the point falls outside the convex hull, the triangulation is restored by flips. Apart from the location, insertion takes a -time \ccTexHtml{$O(1)$}{O(1)}. This bound is only an amortized bound +time $O(1)$. This bound is only an amortized bound for points located outside the convex hull. Removal of a vertex is done by removing all adjacent triangles, and re-triangulating the hole. Removal takes a time at most proportional to -\ccTexHtml{$d^2$}{d^2}, where - \ccTexHtml{$d$}{d} is the degree of the removed vertex, -which is \ccTexHtml{$O(1)$}{O(1)} for a random vertex. +$d^2$, where +$d$ is the degree of the removed vertex, +which is $O(1)$ for a random vertex. The face, edge, and vertex iterators on finite features are derived from their counterparts visiting all (finite and infinite) @@ -624,23 +624,23 @@ The insertion of a new point in the Delaunay triangulation is performed using first the insertion member function of the basic triangulation and second performing a sequence of flips to restore the Delaunay property. -The number of flips that have to be performed is \ccTexHtml{$O(d)$}{O(d)} -if the new vertex has degree \ccTexHtml{$d$}{d} in the updated +The number of flips that have to be performed is $O(d)$ +if the new vertex has degree $d$ in the updated Delaunay triangulation. For points distributed uniformly at random, -each insertion takes time \ccTexHtml{$O(1)$}{O(1)} on +each insertion takes time $O(1)$ on average, once the point has been located in the triangulation. Removal calls the removal in the triangulation and then re-triangulates the hole created in such a way that the Delaunay criterion is satisfied. Removal of a -vertex of degree \ccTexHtml{$d$}{d} takes time \ccTexHtml{$O(d^2)$}{O(d^2)}. -The degree $d$ is \ccTexHtml{$O(1)$}{O(1)} for a random +vertex of degree $d$ takes time $O(d^2)$. +The degree $d$ is $O(1)$ for a random vertex in the triangulation. After having performed a point location, the -nearest neighbor of a point is found in time \ccTexHtml{$O(n)$}{O(n)} in the -worst case, but in time \ccTexHtml{$O(1)$}{O(1)} +nearest neighbor of a point is found in time $O(n)$ in the +worst case, but in time $O(1)$ for vertices distributed uniformly at random and any query point. diff --git a/Triangulation_2/doc_tex/Triangulation_2_ref/Delaunay_triangulation_2.tex b/Triangulation_2/doc_tex/Triangulation_2_ref/Delaunay_triangulation_2.tex index 160d5eb61fe..5849eed95c4 100644 --- a/Triangulation_2/doc_tex/Triangulation_2_ref/Delaunay_triangulation_2.tex +++ b/Triangulation_2/doc_tex/Triangulation_2_ref/Delaunay_triangulation_2.tex @@ -265,20 +265,20 @@ and additionally tests the Delaunay property. This method is \ccHeading{Implementation} Insertion is implemented by inserting in the triangulation, then -performing a sequence of Delaunay flips. The number of flips is \ccTexHtml{$O(d)$}{O(d)} -if the new vertex is of degree \ccTexHtml{$d$}{d} in the new triangulation. For -points distributed uniformly at random, insertion takes time \ccTexHtml{$O(1)$}{O(1)} on +performing a sequence of Delaunay flips. The number of flips is $O(d)$ +if the new vertex is of degree $d$ in the new triangulation. For +points distributed uniformly at random, insertion takes time $O(1)$ on average. Removal calls the removal in the triangulation and then re-triangulates the hole in such a way that the Delaunay criterion is satisfied. Removal of a -vertex of degree \ccTexHtml{$d$}{d} takes time \ccTexHtml{$O(d^2)$}{O(d^2)}. -The degree \ccTexHtml{$d$}{d} is \ccTexHtml{$O(1)$}{O(1)} for a random +vertex of degree $d$ takes time $O(d^2)$. +The degree $d$ is $O(1)$ for a random vertex in the triangulation. After a point location step, the nearest neighbor -is found in time \ccTexHtml{$O(n)$}{O(n)} in the -worst case, but in time \ccTexHtml{$O(1)$}{O(1)} +is found in time $O(n)$ in the +worst case, but in time $O(1)$ for vertices distributed uniformly at random and any query point. diff --git a/Triangulation_2/doc_tex/Triangulation_2_ref/Triangulation_2.tex b/Triangulation_2/doc_tex/Triangulation_2_ref/Triangulation_2.tex index 704f6edce99..0093af6fe40 100644 --- a/Triangulation_2/doc_tex/Triangulation_2_ref/Triangulation_2.tex +++ b/Triangulation_2/doc_tex/Triangulation_2_ref/Triangulation_2.tex @@ -976,20 +976,20 @@ See the \ccc{Qt_widget} class. Locate is implemented by a line walk from a vertex of the face given as optional parameter (or from a finite vertex of \ccStyle{infinite_face()} if no optional parameter is given). It takes -time \ccTexHtml{$O(n)$}{O(n)} in the worst case, but only \ccTexHtml{$O(\sqrt{n})$}{O(sqrt(n))} +time $O(n)$ in the worst case, but only $O(\sqrt{n})$ on average if the vertices are distributed uniformly at random. Insertion of a point is done by locating a face that contains the point, and then splitting this face. If the point falls outside the convex hull, the triangulation is restored by flips. Apart from the location, insertion takes a time -time \ccTexHtml{$O(1)$}{O(1)}. This bound is only an amortized bound +time $O(1)$. This bound is only an amortized bound for points located outside the convex hull. Removal of a vertex is done by removing all adjacent triangles, and -re-triangulating the hole. Removal takes time \ccTexHtml{$O(d^2)$}{O(d^2)} in the worst -case, if \ccTexHtml{$d$}{d} is the degree of the removed vertex, -which is \ccTexHtml{$O(1)$}{O(1)} for a random vertex. +re-triangulating the hole. Removal takes time $O(d^2)$ in the worst +case, if $d$ is the degree of the removed vertex, +which is $O(1)$ for a random vertex. The face, edge, and vertex iterators on finite features are derived from their counterparts visiting all (finite and infinite)