mirror of https://github.com/CGAL/cgal
first time
This commit is contained in:
Eli Packer
parent
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@article{o-naler-90
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, author = "M. Orlowski"
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, title = "A new algorithm for the largest empty rectangle problem"
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, journal = "Algorithmica"
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, volume = 5
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, year = 1990
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, pages = "65--73"
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, keywords = "area"
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, annote = "$O(n \log n)$ expected time, $O(n)$ space"
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}
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% +------------------------------------------------------------------------+
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% | CGAL Reference Manual: snapRounding.tex
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% +------------------------------------------------------------------------+
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% | snap rounding of line segments
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% |
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% | 9.4.00 Eli Packer
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% |
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%\RCSdef{\largestEmptyRectangleRev}{$Revision$}
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%\RCSdefDate{largestEmptyRectangleDate}{$Date$}
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% +------------------------------------------------------------------------+
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\ccParDims
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% \usepackage{graphics, amssymb,epsfig}
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\chapter{Largest Empty Rectangle}
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\label{chapterLer}
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%\ccChapterRelease{\largestEmptyRectangleRev. \ \largestEmptyRectangleDate}\\
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\ccChapterAuthor{Eli Packer}
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% +------------------------------------------------------------------------+
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\section{Overview}
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The Largest Empty Rectangle problem answers the following query. Given
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a set of points in the plane and a bounding box( iso-rectangle) that bounds them,
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find the iso-rectangle with the largest area among all iso-rectangles that are
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inside the above bounding box and do not contain any point of the point set.
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See Figure~\ref{fig:ler1} for an illustration.
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\begin{figure}[h]
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\begin{ccTexOnly}
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\centerline{
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\includegraphics{ler_example.ps}
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}
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\end{ccTexOnly}
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\caption{An example of the largest empty iso rectangle of a set of points
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\label{fig:ler1}}
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\begin{ccHtmlOnly}
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<P>
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<center>
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<img src="ler_example.gif" border=0 alt="example output">
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<!--An example of the largest empty iso rectangle of a set of points-->
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</center>
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\end{ccHtmlOnly}
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\end{figure}
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The algorithm is an implementation of \cite{o-naler-90}. The runtime of an
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insertion or a removal is $O(\log n)$. A query takes $O(n^2)$ worst
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case time and $O(n \log n)$ expected time. The working storage is $
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O(n)$.
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% +========================================================================+
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\section{Examples of Largest Empty Rectangle}
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% +========================================================================+
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The following example generates the Largest Empty Rectangle of a set
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of points.
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\ccIncludeExampleCode{../../examples/Largest_empty_rect_2/example.C}
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% +--------------------------------------------------------+
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% EOF
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@ -0,0 +1,107 @@
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% +------------------------------------------------------------------------+
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% | Reference manual page: LargestEmptyIsoRectangleTraits_2.tex
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% +------------------------------------------------------------------------+
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% | 06.04.2000 Author
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% | Package: Package
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% |
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\RCSdef{\RCSLargestEmptyIsoRectangleTraits_2Rev}{$Revision$}
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\RCSdefDate{\RCSLargestEmptyIsoRectangleTraits_2Date}{$Date$}
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% |
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%%RefPage: end of header, begin of main body
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% +------------------------------------------------------------------------+
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\begin{ccRefConcept}{LargestEmptyIsoRectangleTraits_2}
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%% \ccHtmlCrossLink{} %% add further rules for cross referencing links
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%% \ccHtmlIndexC[concept]{} %% add further index entries
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\ccDefinition
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The concept \ccRefName\ describes the set of requirements to be
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fulfilled by any class used to instantiate the template parameter of
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the class \ccc{Largest_empty_iso_rectangle_2<T>}.
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This concept provides the types of the geometric primitives used in
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this class and some function object types for the required
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predicates on those primitives.
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%% Less_x_2, Less_yx_2, Less_yx_2, Compare_x_2, Compare_y_2, Comparison_result
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\ccTypes
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\ccNestedType{Point_2}{The point type.}
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\ccGlue
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\ccNestedType{Iso_rectangle_2}{The iso rectangle type.}
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\ccNestedType{Compare_x_2_object}{Predicate object. Must provide
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the operator
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\ccc{Comparison_result operator()(Point_2 p, Point_2 q)}
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which returns
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\ccc{SMALLER, EQUAL} or \ccc{ LARGER}
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according ding to the
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$x$-ordering of points \ccc{p} and \ccc{q}.}
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\ccGlue
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\ccNestedType{Compare_y_2_object}{Predicate object. Must provide
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the operator
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\ccc{Comparison_result operator()(Point_2 p, Point_2 q)}
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which returns
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\ccc{SMALLER, EQUAL} or \ccc{ LARGER}
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according to the
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$y$-ordering of points \ccc{p} and \ccc{q}.}
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\ccGlue
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\ccNestedType{Less_x_2_object}{Predicate object. Must provide
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the operator
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\ccc{bool operator()(Point_2 p, Point_2 q)}
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which returns
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whether \ccc{p} is less than \ccc{q} according to their $x$-ordering.}
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\ccGlue
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\ccNestedType{Less_y_2_object}{Predicate object. Must provide
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the operator
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\ccc{bool operator()(Point_2 p, Point_2 q)}
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which returns
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whether \ccc{p} is less than \ccc{q} according to their $y$-ordering.}
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\ccCreation
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\ccCreationVariable{traits} %% choose variable name
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Only a default constructor, copy constructor
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and an assignement operator are required.
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Note that further constructors
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can be provided.
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\ccConstructor{LargestEmptyIsoRectangleTraits_2();}{Default constructor.}
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\ccGlue
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\ccConstructor{LargestEmptyIsoRectangleTraits_2(LargestEmptyIsoRectangleTraits_2);}
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{Copy constructor}
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\ccMethod{LargestEmptyIsoRectangleTraits_2 operator=(LargestEmptyIsoRectangleTraits_2 gtr);}
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{Assignment operator.}
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\ccHeading{Predicate functions}
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The following functions give access to the predicate
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and constructor objects.
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\ccThree{Construct_segment_2}{gt.compare_x(Point p0, Point p1)x}{}
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\ccMethod{Comparison_x_2 compare_x_2_object();}{}
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\ccGlue
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\ccMethod{Comparison_y_2 compare_y_2_object();}{}
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\ccGlue
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\ccMethod{bool less_x_2_object();}{}
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\ccGlue
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\ccMethod{bool less_y_2_object();}{}
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\ccHasModels
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\ccc{CGAL::Cartesian<R>} \\
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\ccc{CGAL::Homogeneous<R>}
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\ccSeeAlso
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\ccc{CGAL::Largest_empty_iso_rectangle_2<Traits>}
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\end{ccRefConcept}
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% +------------------------------------------------------------------------+
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%%RefPage: end of main body, begin of footer
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% EOF
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% +------------------------------------------------------------------------+
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@ -0,0 +1,11 @@
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@article{o-naler-90
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, author = "M. Orlowski"
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, title = "A new algorithm for the largest empty rectangle problem"
|
||||
, journal = "Algorithmica"
|
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, volume = 5
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, year = 1990
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, pages = "65--73"
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, keywords = "area"
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, annote = "$O(n \log n)$ expected time, $O(n)$ space"
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}
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@ -0,0 +1,156 @@
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% +------------------------------------------------------------------------+
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% | Reference manual page: Largest_empty_iso_rectangle_2.tex
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% +------------------------------------------------------------------------+
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% | 27.3.2000 Eli Packer
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% | Package:
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% |
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%\RCSdef{\RCSTriangulationRev}{$Revision$}
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%\RCSdefDate{\RCSTriangulationDate}{$Date$}
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% |
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%%RefPage: end of header, begin of main body
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% +------------------------------------------------------------------------+
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\begin{ccRefClass}{Largest_empty_iso_rectangle_2<T>}
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%% \ccHtmlCrossLink{} %% add further rules for cross referencing links
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%% \ccHtmlIndexC[class]{} %% add further index entries
|
||||
|
||||
|
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\ccDefinition
|
||||
|
||||
Given a set of points in the plane, the class \ccRefName\ is a data
|
||||
structure that maintains an iso-rectangle with the largest area among
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||||
all iso-rectangles that are inside a given bounding box( iso-rectangle), and
|
||||
that do not contain any point of the point set.
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||||
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||||
The class \ccRefName\ expects a model of the concept \ccc{LargestEmptyIsoRectangleTraits_2} as its template argument.
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||||
\ccInclude{CGAL/Largest_empty_iso_rectangle_2.h}
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||||
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\ccTypes
|
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The class \ccClassTemplateName\ defines the following types:
|
||||
|
||||
\ccThreeToTwo
|
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||||
\ccTypedef{typedef T Traits;}{}
|
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|
||||
\ccTypedef{typedef Traits::Point_2 Point_2;}{}
|
||||
|
||||
\ccTypedef{typedef Traits::Iso_rectangle_2 Iso_rectangle_2;}{}
|
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|
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The following iterator allows to enumerate the points.
|
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It is non mutable, bidirectional
|
||||
and its value type is \ccc{Point_2}.
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It is invalidated by any insertion or removal of a point.
|
||||
|
||||
\ccNestedType{const_iterator}{Iterator over the points.}
|
||||
|
||||
|
||||
\ccCreation
|
||||
\ccCreationVariable{l} %% choose variable name
|
||||
\ccSetTwoColumns{Qt_widget}{}
|
||||
%\ccThree{Largest_empty_iso_rectangle_2<Traits>(const Point_2& bl, const Point_2& tr)}{}{}
|
||||
|
||||
\ccConstructor{Largest_empty_iso_rectangle_2<Traits>
|
||||
(const Iso_rectangle_2 &b);}
|
||||
{Constructor. The iso-rectangle \ccc{b} is the bounding rectangle.}
|
||||
|
||||
\ccConstructor{Largest_empty_iso_rectangle_2<Traits>
|
||||
(const Point_2 p,const Point_2 q);}
|
||||
{Constructor. The iso-rectangle whose lower left and upper right points are \ccc{p} and
|
||||
\ccc{q} respectively is the bounding rectangle.}
|
||||
|
||||
\ccConstructor{Largest_empty_iso_rectangle_2<Traits>
|
||||
();}
|
||||
{Constructor. The iso-rectangle whose lower left point and upper right points are (0,0)
|
||||
and (1,1) respectively is the bounding rectangle.}
|
||||
|
||||
\ccConstructor{Largest_empty_iso_rectangle_2<Traits>
|
||||
(const Largest_empty_iso_rectangle_2<Traits> tr);}
|
||||
{Copy constructor.}
|
||||
|
||||
%\ccConstructor{\tilde Largest_empty_iso_rectangle_2<Traits>();}
|
||||
%{Destructor.}
|
||||
%
|
||||
\ccOperations
|
||||
\ccSetThreeColumns{const_iterator}{container.begin() const;}{}
|
||||
|
||||
\ccHeading{Assignment}
|
||||
|
||||
\ccMethod{Largest_empty_iso_rectangle_2<T>
|
||||
operator=(const Largest_empty_iso_rectangle_2<T> & tr);}
|
||||
{}
|
||||
|
||||
\ccAccessFunctions
|
||||
|
||||
\ccMethod{const Traits & traits() const;}
|
||||
{Returns a const reference to the traits object.}
|
||||
|
||||
|
||||
\ccMethod{const_iterator begin() const;}
|
||||
{Returns an iterator to the beginning of the point set.}
|
||||
\ccMethod{const_iterator end() const;}
|
||||
{Returns a past-the-end iterator for the point set.}
|
||||
|
||||
|
||||
\ccHeading{Queries}
|
||||
|
||||
\ccMethod{Quadruple<Point_2, Point_2, Point_2, Point_2>
|
||||
get_left_bottom_right_top();}
|
||||
{Returns the four points that define the largest empty iso-rectangle.
|
||||
(Note that these points are not necessarily on a corner of an iso-rectangle.)}
|
||||
\ccGlue
|
||||
\ccMethod{Iso_rectangle_2 get_largest_empty_iso_rectangle();}
|
||||
{Returns the largest empty iso-rectangle. (Note that the two
|
||||
points defining the iso-rectangle are not necessarily part of
|
||||
the point set.)}
|
||||
\ccGlue
|
||||
\ccMethod{Iso_rectangle_2 get_bounding_box();}
|
||||
{Returns the iso-rectangle passed in the constructor.}
|
||||
\ccHeading{Insertion}
|
||||
|
||||
\ccMethod{void
|
||||
insert(const Point_2& p);}
|
||||
{Inserts point \ccc{p} in the point set, if it is not already in the set.}
|
||||
|
||||
\ccMethod{void
|
||||
push_back(const Point_2& p);}
|
||||
{Inserts point \ccc{p} in the point set, if it is not already in the set.}
|
||||
|
||||
\ccMethod{template < class InputIterator >
|
||||
int
|
||||
insert(InputIterator first, InputIterator last);}
|
||||
{Inserts the points in the range $\left[\right.$\ccc{first},
|
||||
\ccc{last}$\left.\right)$. Returns the number of inserted points. \\ \\
|
||||
\ccRequirements The \ccc{value_type} of \ccc{first} and \ccc{last} is \ccc{Point}.}
|
||||
|
||||
\ccHeading{Removal}
|
||||
|
||||
\ccMethod{bool remove(const Point_2& p);}{Removes point \ccc{p}.
|
||||
Returns false iff \ccc{p} is not in the point set. }
|
||||
|
||||
\ccMethod{void clear();}
|
||||
{Removes all points of \ccVar.}
|
||||
|
||||
%\ccSeeAlso
|
||||
|
||||
\ccImplementation
|
||||
|
||||
The algorithm is an implementation of \cite{o-naler-90}. The runtime of an
|
||||
insertion or a removal is $O(\log n)$. A query takes $O(n^2)$ worst
|
||||
case time and $O(n \log n)$ expected time. The working storage is $
|
||||
O(n)$.
|
||||
|
||||
\bibliography{Largest_empty_iso_rectangle_2}
|
||||
\bibliographystyle{abbrv}
|
||||
|
||||
\end{ccRefClass}
|
||||
|
||||
% +------------------------------------------------------------------------+
|
||||
%%RefPage: end of main body, begin of footer
|
||||
% EOF
|
||||
% +------------------------------------------------------------------------+
|
||||
|
||||
|
|
@ -0,0 +1,11 @@
|
|||
|
||||
@article{o-naler-90
|
||||
, author = "M. Orlowski"
|
||||
, title = "A new algorithm for the largest empty rectangle problem"
|
||||
, journal = "Algorithmica"
|
||||
, volume = 5
|
||||
, year = 1990
|
||||
, pages = "65--73"
|
||||
, keywords = "area"
|
||||
, annote = "$O(n \log n)$ expected time, $O(n)$ space"
|
||||
}
|
||||
|
|
@ -0,0 +1,67 @@
|
|||
% +------------------------------------------------------------------------+
|
||||
% | CGAL Reference Manual: snapRounding.tex
|
||||
% +------------------------------------------------------------------------+
|
||||
% | snap rounding of line segments
|
||||
% |
|
||||
% | 9.4.00 Eli Packer
|
||||
% |
|
||||
%\RCSdef{\largestEmptyRectangleRev}{$Revision$}
|
||||
%\RCSdefDate{largestEmptyRectangleDate}{$Date$}
|
||||
% +------------------------------------------------------------------------+
|
||||
|
||||
\ccParDims
|
||||
|
||||
% \usepackage{graphics, amssymb,epsfig}
|
||||
|
||||
\chapter{Largest Empty Rectangle}
|
||||
\label{chapterLer}
|
||||
%\ccChapterRelease{\largestEmptyRectangleRev. \ \largestEmptyRectangleDate}\\
|
||||
\ccChapterAuthor{Eli Packer}
|
||||
|
||||
% +------------------------------------------------------------------------+
|
||||
\section{Overview}
|
||||
The Largest Empty Rectangle problem answers the following query. Given
|
||||
a set of points in the plane and a bounding box( iso-rectangle) that bounds them,
|
||||
find the iso-rectangle with the largest area among all iso-rectangles that are
|
||||
inside the above bounding box and do not contain any point of the point set.
|
||||
See Figure~\ref{fig:ler1} for an illustration.
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{ccTexOnly}
|
||||
\centerline{
|
||||
\includegraphics{ler_example.ps}
|
||||
}
|
||||
\end{ccTexOnly}
|
||||
|
||||
\caption{An example of the largest empty iso rectangle of a set of points
|
||||
\label{fig:ler1}}
|
||||
|
||||
\begin{ccHtmlOnly}
|
||||
<P>
|
||||
<center>
|
||||
<img src="ler_example.gif" border=0 alt="example output">
|
||||
<!--An example of the largest empty iso rectangle of a set of points-->
|
||||
</center>
|
||||
\end{ccHtmlOnly}
|
||||
\end{figure}
|
||||
|
||||
The algorithm is an implementation of \cite{o-naler-90}. The runtime of an
|
||||
insertion or a removal is $O(\log n)$. A query takes $O(n^2)$ worst
|
||||
case time and $O(n \log n)$ expected time. The working storage is $
|
||||
O(n)$.
|
||||
|
||||
|
||||
% +========================================================================+
|
||||
\section{Examples of Largest Empty Rectangle}
|
||||
% +========================================================================+
|
||||
|
||||
The following example generates the Largest Empty Rectangle of a set
|
||||
of points.
|
||||
|
||||
\ccIncludeExampleCode{../../examples/Largest_empty_rect_2/example.C}
|
||||
|
||||
% +--------------------------------------------------------+
|
||||
|
||||
% EOF
|
||||
|
||||
|
||||
|
|
@ -0,0 +1,200 @@
|
|||
%!PS-Adobe-2.0
|
||||
%%Title: ler_example.ps
|
||||
%%Creator: fig2dev Version 3.2.3 Patchlevel
|
||||
%%CreationDate: Sun Feb 17 21:28:07 2002
|
||||
%%For: elip@nova (Eli Packer,,036722528,)
|
||||
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|
||||
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||||
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||||
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||||
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90 rotate
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1 -1 scale
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4 -2 roll dup 1 exch sub 3 -1 roll mul add
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4 -2 roll dup 1 exch sub 3 -1 roll mul add
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4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
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bind def
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/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
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/DrawEllipse {
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/endangle exch def
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x y tr xrad yrad sc 0 0 1 startangle endangle arc
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closepath
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savematrix setmatrix
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} def
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/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
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$F2psBegin
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% Ellipse
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% Ellipse
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% Ellipse
|
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n 9195 5400 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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||||
% Ellipse
|
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n 6662 2191 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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||||
% Ellipse
|
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n 7110 3195 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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||||
% Ellipse
|
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% Ellipse
|
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n 8055 4170 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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% Ellipse
|
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n 5970 7170 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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% Ellipse
|
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n 7110 6645 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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% Ellipse
|
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% Ellipse
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|
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n 9780 1575 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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% Ellipse
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n 7980 2100 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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|
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n 1910 5155 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
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||||
% Ellipse
|
||||
n 3675 5715 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
|
||||
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||||
% Polyline
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7.500 slw
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cp gs col0 s gr
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% Polyline
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30.000 slw
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$F2psEnd
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rs
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showpage
|
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|
|
@ -0,0 +1,107 @@
|
|||
% +------------------------------------------------------------------------+
|
||||
% | Reference manual page: LargestEmptyIsoRectangleTraits_2.tex
|
||||
% +------------------------------------------------------------------------+
|
||||
% | 06.04.2000 Author
|
||||
% | Package: Package
|
||||
% |
|
||||
\RCSdef{\RCSLargestEmptyIsoRectangleTraits_2Rev}{$Revision$}
|
||||
\RCSdefDate{\RCSLargestEmptyIsoRectangleTraits_2Date}{$Date$}
|
||||
% |
|
||||
%%RefPage: end of header, begin of main body
|
||||
% +------------------------------------------------------------------------+
|
||||
|
||||
|
||||
\begin{ccRefConcept}{LargestEmptyIsoRectangleTraits_2}
|
||||
|
||||
%% \ccHtmlCrossLink{} %% add further rules for cross referencing links
|
||||
%% \ccHtmlIndexC[concept]{} %% add further index entries
|
||||
|
||||
\ccDefinition
|
||||
|
||||
The concept \ccRefName\ describes the set of requirements to be
|
||||
fulfilled by any class used to instantiate the template parameter of
|
||||
the class \ccc{Largest_empty_iso_rectangle_2<T>}.
|
||||
This concept provides the types of the geometric primitives used in
|
||||
this class and some function object types for the required
|
||||
predicates on those primitives.
|
||||
|
||||
%% Less_x_2, Less_yx_2, Less_yx_2, Compare_x_2, Compare_y_2, Comparison_result
|
||||
|
||||
\ccTypes
|
||||
\ccNestedType{Point_2}{The point type.}
|
||||
\ccGlue
|
||||
\ccNestedType{Iso_rectangle_2}{The iso rectangle type.}
|
||||
|
||||
\ccNestedType{Compare_x_2_object}{Predicate object. Must provide
|
||||
the operator
|
||||
\ccc{Comparison_result operator()(Point_2 p, Point_2 q)}
|
||||
which returns
|
||||
\ccc{SMALLER, EQUAL} or \ccc{ LARGER}
|
||||
according ding to the
|
||||
$x$-ordering of points \ccc{p} and \ccc{q}.}
|
||||
\ccGlue
|
||||
\ccNestedType{Compare_y_2_object}{Predicate object. Must provide
|
||||
the operator
|
||||
\ccc{Comparison_result operator()(Point_2 p, Point_2 q)}
|
||||
which returns
|
||||
\ccc{SMALLER, EQUAL} or \ccc{ LARGER}
|
||||
according to the
|
||||
$y$-ordering of points \ccc{p} and \ccc{q}.}
|
||||
\ccGlue
|
||||
\ccNestedType{Less_x_2_object}{Predicate object. Must provide
|
||||
the operator
|
||||
\ccc{bool operator()(Point_2 p, Point_2 q)}
|
||||
which returns
|
||||
whether \ccc{p} is less than \ccc{q} according to their $x$-ordering.}
|
||||
\ccGlue
|
||||
\ccNestedType{Less_y_2_object}{Predicate object. Must provide
|
||||
the operator
|
||||
\ccc{bool operator()(Point_2 p, Point_2 q)}
|
||||
which returns
|
||||
whether \ccc{p} is less than \ccc{q} according to their $y$-ordering.}
|
||||
|
||||
\ccCreation
|
||||
\ccCreationVariable{traits} %% choose variable name
|
||||
Only a default constructor, copy constructor
|
||||
and an assignement operator are required.
|
||||
Note that further constructors
|
||||
can be provided.
|
||||
|
||||
\ccConstructor{LargestEmptyIsoRectangleTraits_2();}{Default constructor.}
|
||||
\ccGlue
|
||||
\ccConstructor{LargestEmptyIsoRectangleTraits_2(LargestEmptyIsoRectangleTraits_2);}
|
||||
{Copy constructor}
|
||||
\ccMethod{LargestEmptyIsoRectangleTraits_2 operator=(LargestEmptyIsoRectangleTraits_2 gtr);}
|
||||
{Assignment operator.}
|
||||
|
||||
\ccHeading{Predicate functions}
|
||||
|
||||
The following functions give access to the predicate
|
||||
and constructor objects.
|
||||
|
||||
\ccThree{Construct_segment_2}{gt.compare_x(Point p0, Point p1)x}{}
|
||||
|
||||
\ccMethod{Comparison_x_2 compare_x_2_object();}{}
|
||||
\ccGlue
|
||||
\ccMethod{Comparison_y_2 compare_y_2_object();}{}
|
||||
\ccGlue
|
||||
\ccMethod{bool less_x_2_object();}{}
|
||||
\ccGlue
|
||||
\ccMethod{bool less_y_2_object();}{}
|
||||
|
||||
\ccHasModels
|
||||
\ccc{CGAL::Cartesian<R>} \\
|
||||
\ccc{CGAL::Homogeneous<R>}
|
||||
|
||||
\ccSeeAlso
|
||||
\ccc{CGAL::Largest_empty_iso_rectangle_2<Traits>}
|
||||
|
||||
|
||||
|
||||
\end{ccRefConcept}
|
||||
|
||||
% +------------------------------------------------------------------------+
|
||||
%%RefPage: end of main body, begin of footer
|
||||
% EOF
|
||||
% +------------------------------------------------------------------------+
|
||||
|
||||
|
|
@ -0,0 +1,11 @@
|
|||
|
||||
@article{o-naler-90
|
||||
, author = "M. Orlowski"
|
||||
, title = "A new algorithm for the largest empty rectangle problem"
|
||||
, journal = "Algorithmica"
|
||||
, volume = 5
|
||||
, year = 1990
|
||||
, pages = "65--73"
|
||||
, keywords = "area"
|
||||
, annote = "$O(n \log n)$ expected time, $O(n)$ space"
|
||||
}
|
||||
|
|
@ -0,0 +1,156 @@
|
|||
% +------------------------------------------------------------------------+
|
||||
% | Reference manual page: Largest_empty_iso_rectangle_2.tex
|
||||
% +------------------------------------------------------------------------+
|
||||
% | 27.3.2000 Eli Packer
|
||||
% | Package:
|
||||
% |
|
||||
%\RCSdef{\RCSTriangulationRev}{$Revision$}
|
||||
%\RCSdefDate{\RCSTriangulationDate}{$Date$}
|
||||
% |
|
||||
%%RefPage: end of header, begin of main body
|
||||
% +------------------------------------------------------------------------+
|
||||
|
||||
|
||||
\begin{ccRefClass}{Largest_empty_iso_rectangle_2<T>}
|
||||
|
||||
%% \ccHtmlCrossLink{} %% add further rules for cross referencing links
|
||||
%% \ccHtmlIndexC[class]{} %% add further index entries
|
||||
|
||||
|
||||
\ccDefinition
|
||||
|
||||
Given a set of points in the plane, the class \ccRefName\ is a data
|
||||
structure that maintains an iso-rectangle with the largest area among
|
||||
all iso-rectangles that are inside a given bounding box( iso-rectangle), and
|
||||
that do not contain any point of the point set.
|
||||
|
||||
The class \ccRefName\ expects a model of the concept \ccc{LargestEmptyIsoRectangleTraits_2} as its template argument.
|
||||
|
||||
\ccInclude{CGAL/Largest_empty_iso_rectangle_2.h}
|
||||
|
||||
|
||||
\ccTypes
|
||||
The class \ccClassTemplateName\ defines the following types:
|
||||
|
||||
\ccThreeToTwo
|
||||
|
||||
\ccTypedef{typedef T Traits;}{}
|
||||
|
||||
\ccTypedef{typedef Traits::Point_2 Point_2;}{}
|
||||
|
||||
\ccTypedef{typedef Traits::Iso_rectangle_2 Iso_rectangle_2;}{}
|
||||
|
||||
|
||||
The following iterator allows to enumerate the points.
|
||||
It is non mutable, bidirectional
|
||||
and its value type is \ccc{Point_2}.
|
||||
It is invalidated by any insertion or removal of a point.
|
||||
|
||||
\ccNestedType{const_iterator}{Iterator over the points.}
|
||||
|
||||
|
||||
\ccCreation
|
||||
\ccCreationVariable{l} %% choose variable name
|
||||
\ccSetTwoColumns{Qt_widget}{}
|
||||
%\ccThree{Largest_empty_iso_rectangle_2<Traits>(const Point_2& bl, const Point_2& tr)}{}{}
|
||||
|
||||
\ccConstructor{Largest_empty_iso_rectangle_2<Traits>
|
||||
(const Iso_rectangle_2 &b);}
|
||||
{Constructor. The iso-rectangle \ccc{b} is the bounding rectangle.}
|
||||
|
||||
\ccConstructor{Largest_empty_iso_rectangle_2<Traits>
|
||||
(const Point_2 p,const Point_2 q);}
|
||||
{Constructor. The iso-rectangle whose lower left and upper right points are \ccc{p} and
|
||||
\ccc{q} respectively is the bounding rectangle.}
|
||||
|
||||
\ccConstructor{Largest_empty_iso_rectangle_2<Traits>
|
||||
();}
|
||||
{Constructor. The iso-rectangle whose lower left point and upper right points are (0,0)
|
||||
and (1,1) respectively is the bounding rectangle.}
|
||||
|
||||
\ccConstructor{Largest_empty_iso_rectangle_2<Traits>
|
||||
(const Largest_empty_iso_rectangle_2<Traits> tr);}
|
||||
{Copy constructor.}
|
||||
|
||||
%\ccConstructor{\tilde Largest_empty_iso_rectangle_2<Traits>();}
|
||||
%{Destructor.}
|
||||
%
|
||||
\ccOperations
|
||||
\ccSetThreeColumns{const_iterator}{container.begin() const;}{}
|
||||
|
||||
\ccHeading{Assignment}
|
||||
|
||||
\ccMethod{Largest_empty_iso_rectangle_2<T>
|
||||
operator=(const Largest_empty_iso_rectangle_2<T> & tr);}
|
||||
{}
|
||||
|
||||
\ccAccessFunctions
|
||||
|
||||
\ccMethod{const Traits & traits() const;}
|
||||
{Returns a const reference to the traits object.}
|
||||
|
||||
|
||||
\ccMethod{const_iterator begin() const;}
|
||||
{Returns an iterator to the beginning of the point set.}
|
||||
\ccMethod{const_iterator end() const;}
|
||||
{Returns a past-the-end iterator for the point set.}
|
||||
|
||||
|
||||
\ccHeading{Queries}
|
||||
|
||||
\ccMethod{Quadruple<Point_2, Point_2, Point_2, Point_2>
|
||||
get_left_bottom_right_top();}
|
||||
{Returns the four points that define the largest empty iso-rectangle.
|
||||
(Note that these points are not necessarily on a corner of an iso-rectangle.)}
|
||||
\ccGlue
|
||||
\ccMethod{Iso_rectangle_2 get_largest_empty_iso_rectangle();}
|
||||
{Returns the largest empty iso-rectangle. (Note that the two
|
||||
points defining the iso-rectangle are not necessarily part of
|
||||
the point set.)}
|
||||
\ccGlue
|
||||
\ccMethod{Iso_rectangle_2 get_bounding_box();}
|
||||
{Returns the iso-rectangle passed in the constructor.}
|
||||
\ccHeading{Insertion}
|
||||
|
||||
\ccMethod{void
|
||||
insert(const Point_2& p);}
|
||||
{Inserts point \ccc{p} in the point set, if it is not already in the set.}
|
||||
|
||||
\ccMethod{void
|
||||
push_back(const Point_2& p);}
|
||||
{Inserts point \ccc{p} in the point set, if it is not already in the set.}
|
||||
|
||||
\ccMethod{template < class InputIterator >
|
||||
int
|
||||
insert(InputIterator first, InputIterator last);}
|
||||
{Inserts the points in the range $\left[\right.$\ccc{first},
|
||||
\ccc{last}$\left.\right)$. Returns the number of inserted points. \\ \\
|
||||
\ccRequirements The \ccc{value_type} of \ccc{first} and \ccc{last} is \ccc{Point}.}
|
||||
|
||||
\ccHeading{Removal}
|
||||
|
||||
\ccMethod{bool remove(const Point_2& p);}{Removes point \ccc{p}.
|
||||
Returns false iff \ccc{p} is not in the point set. }
|
||||
|
||||
\ccMethod{void clear();}
|
||||
{Removes all points of \ccVar.}
|
||||
|
||||
%\ccSeeAlso
|
||||
|
||||
\ccImplementation
|
||||
|
||||
The algorithm is an implementation of \cite{o-naler-90}. The runtime of an
|
||||
insertion or a removal is $O(\log n)$. A query takes $O(n^2)$ worst
|
||||
case time and $O(n \log n)$ expected time. The working storage is $
|
||||
O(n)$.
|
||||
|
||||
\bibliography{Largest_empty_iso_rectangle_2}
|
||||
\bibliographystyle{abbrv}
|
||||
|
||||
\end{ccRefClass}
|
||||
|
||||
% +------------------------------------------------------------------------+
|
||||
%%RefPage: end of main body, begin of footer
|
||||
% EOF
|
||||
% +------------------------------------------------------------------------+
|
||||
|
||||
Loading…
Reference in New Issue