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<doc>
   <title>Algorithms</title>

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   <body>
      <br/><br/>

         <p>
            This page documents library components that are all basically just implementations of 
            mathematical functions or algorithms that don't fit in any of the other pages
            of the dlib documentation.  So this includes things like checksums, cryptographic hashes, 
            sorting, etc.
         </p>

   </body>

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   <menu width="150">
    <top>
      <section>
         <name>Tools</name>
         <item>bigint</item> 
         <item>disjoint_subsets</item> 
         <item nolink="true">
            <name>Quantum Computing</name>
            <sub>
               <item>quantum_register</item> 
               <item>gate</item> 
            </sub>
         </item>
         <item nolink="true">
            <name>Geometry</name>
            <sub>
               <item>border_enumerator</item> 
               <item>rectangle</item> 
               <item>vector</item> 
               <item>point</item> 
               <item>rotate_point</item> 
               <item>point_rotator</item> 
               <item>get_rect</item> 
               <item>centered_rect</item> 
               <item>center</item> 
               <item>shrink_rect</item> 
               <item>grow_rect</item> 
               <item>translate_rect</item> 
               <item>resize_rect</item> 
               <item>resize_rect_width</item> 
               <item>resize_rect_height</item> 
               <item>move_rect</item> 
               <item>nearest_point</item> 
               <item>distance_to_rect_edge</item> 
            </sub>
         </item>
         <item nolink="true">
            <name>Graph Utilities</name>
            <sub>
               <item>graph_contains_directed_cycle</item> 
               <item>graph_contains_undirected_cycle</item> 
               <item>create_moral_graph</item> 
               <item>triangulate_graph_and_find_cliques</item> 
               <item>graph_contains_length_one_cycle</item> 
               <item>find_connected_nodes</item> 
               <item>graph_is_connected</item> 
               <item>is_clique</item> 
               <item>is_maximal_clique</item> 
               <item>copy_graph_structure</item> 
               <item>edge</item> 
               <item>is_join_tree</item> 
               <item>create_join_tree</item> 
            </sub>
         </item>
         <item>hsort_array</item> 
         <item>isort_array</item> 
         <item>put_in_range</item> 
         <item>qsort_array</item> 
         <item>square_root</item> 
         <item nolink="true">
            <name>Set Utilities</name>
            <sub>
               <item>set_intersection_size</item> 
               <item>set_intersection</item> 
               <item>set_union</item> 
               <item>set_difference</item> 
            </sub>
         </item>
      </section>

      <section>
         <name>Statistics</name>
         <item>rand</item> 
         <item>median</item> 
         <item>running_stats</item> 
         <item>running_scalar_covariance</item> 
         <item>mean_sign_agreement</item> 
         <item>correlation</item> 
         <item>covariance</item> 
         <item>r_squared</item> 
         <item>mean_squared_error</item> 
         <item>running_covariance</item> 
         <item>random_subset_selector</item> 
         <item>randomly_subsample</item> 
         <item>randomly_sample_image_features</item> 

      </section>

      <section>
         <name>Hashing</name>
         <item>md5</item> 
         <item>crc32</item> 
         <item>hash</item> 
         <item>murmur_hash3</item> 
         <item>projection_hash</item> 
         <item>create_random_projection_hash</item> 
      </section>
    </top>  
   </menu>

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   <components>
   
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      <component checked="true">
         <name>bigint</name>
         <file>dlib/bigint.h</file>
         <spec_file>dlib/bigint/bigint_kernel_abstract.h</spec_file>
         <description>
            This object represents an arbitrary precision unsigned integer.  It's pretty simple.  
            It's interface is just like a normal int, you don't have to tell it how much memory
            to use or anything unusual.  It just goes :)       
         </description>
         
         <implementations>
            <implementation>
               <name>bigint_kernel_1</name>
               <file>dlib/bigint/bigint_kernel_1.h</file>
               <description> 
                  This implementation is done using an array of unsigned shorts.  It is also reference counted.  
                  For further details see the above link.  Also note that kernel_2 should be 
      faster in almost every case so you should really just use that version of the bigint object.
               </description> 
    
               <typedefs>
                  <typedef>
                     <name>kernel_1a</name>
                     <description>is a typedef for bigint_kernel_1</description>
                  </typedef>
               </typedefs>                
               
            </implementation> 
                     
            <implementation>
               <name>bigint_kernel_2</name>
               <file>dlib/bigint/bigint_kernel_2.h</file>
               <description> 
                  This implementation is basically the same as kernel_1 except it uses the 
                  Fast Fourier Transform to perform multiplications much faster.  
               </description> 
    
               <typedefs>
                  <typedef>
                     <name>kernel_2a</name>
                     <description>is a typedef for bigint_kernel_2</description>
                  </typedef>
               </typedefs>                
               
            </implementation> 
                     
         </implementations>
                           
                        
      </component>
      
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      <component>
         <name>crc32</name>
         <file>dlib/crc32.h</file>
         <spec_file>dlib/crc32/crc32_kernel_abstract.h</spec_file>
         <description>
            This object represents the CRC-32 algorithm for calculating checksums.   
         </description>
         
      </component>
            
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      <component>
         <name>murmur_hash3</name>
         <file>dlib/hash.h</file>
         <spec_file>dlib/general_hash/murmur_hash3_abstract.h</spec_file>
         <description>
            This function takes a block of memory and returns a 32bit hash.  The 
            hashing algorithm used is Austin Appleby's excellent 
            <a href="http://code.google.com/p/smhasher/">MurmurHash3</a>. 
         </description>
         
      </component>
            
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      <component>
         <name>projection_hash</name>
         <file>dlib/lsh.h</file>
         <spec_file>dlib/lsh/projection_hash_abstract.h</spec_file>
         <description>
                This is a tool for hashing elements of a vector space into the integers.  
                It is intended to represent locality sensitive hashing functions such as 
                the popular <a href="#create_random_projection_hash">random projection hashing</a> method.
         </description>
         
      </component>
            
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      <component>
         <name>create_random_projection_hash</name>
         <file>dlib/lsh.h</file>
         <spec_file>dlib/lsh/create_random_projection_hash_abstract.h</spec_file>
         <description>
            Creates a random projection based locality sensitive 
            <a href="#projection_hash">hashing function</a>.
         </description>
         
      </component>
            
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      <component>
         <name>hash</name>
         <file>dlib/hash.h</file>
         <spec_file>dlib/general_hash/hash_abstract.h</spec_file>
         <description>
            This is a set of convenience functions for invoking <a href="#murmur_hash3">murmur_hash3</a>
            on std::strings, std::vectors, std::maps, or <a href="containers.html#matrix">dlib::matrix</a> objects.
            <p>
               As an aside, the hash() for matrix objects is defined <a href="dlib/matrix/matrix_utilities_abstract.h.html#hash">here</a>.
               It has the same interface as all the others.
            </p>
         </description>
         
      </component>
            
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      <component>
         <name>rand</name>
         <file>dlib/rand.h</file>
         <spec_file>dlib/rand/rand_kernel_abstract.h</spec_file>
         <description>
            This object represents a pseudorandom number generator.
         </description>
         
      </component>
            
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      <component>
         <name>rectangle</name>
         <file>dlib/geometry.h</file>
         <spec_file>dlib/geometry/rectangle_abstract.h</spec_file>
         <description>
            This object represents a rectangular region inside a Cartesian 
            coordinate system.  It allows you to easily represent and manipulate
            rectangles.
         </description>
      </component>
            
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      <component>
         <name>border_enumerator</name>
         <file>dlib/geometry.h</file>
         <spec_file>dlib/geometry/border_enumerator_abstract.h</spec_file>
         <description>
            This object is an <a href="containers.html#enumerable">enumerator</a> 
            over the border <a href="#point">points</a> of a 
            <a href="#rectangle">rectangle</a>.
         </description>
      </component>
            
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      <component>
         <name>vector</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/vector_abstract.h</spec_file>
         <description>
            This object represents a two or three dimensional vector.
         </description>
      </component>
      
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      <component>
         <name>point</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/vector_abstract.h</spec_file>
         <description>
               This object represents a point inside a Cartesian coordinate system.
         </description>
      </component>
      
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      <component>
         <name>disjoint_subsets</name>
         <file>dlib/disjoint_subsets.h</file>
         <spec_file link="true">dlib/disjoint_subsets/disjoint_subsets_abstract.h</spec_file>
         <description>
                This object represents a set of integers which is partitioned into
                a number of disjoint subsets.  It supports the two fundamental operations
                of finding which subset a particular integer belongs to as well as 
                merging subsets. 
         </description>

      </component>
      
      
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      <component>
         <name>running_stats</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object represents something that can compute the running mean and
                variance of a stream of real numbers.  
         </description>

         <examples>
            <example>kcentroid_ex.cpp.html</example>
         </examples>
                                 
      </component>
      
      
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      <component>
         <name>running_covariance</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object is a simple tool for computing the mean and
                covariance of a sequence of vectors.  
         </description>

      </component>
      
      
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      <component>
         <name>running_scalar_covariance</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object is a simple tool for computing the covariance of a 
                sequence of scalar values.  
         </description>
      </component>
      
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      <component>
         <name>mean_sign_agreement</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the probability that 
               matching elements of two std::vectors have the same sign.
         </description>
      </component>
      
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      <component>
         <name>correlation</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the correlation between 
               matching elements of two std::vectors.
         </description>
      </component>
      
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      <component>
         <name>covariance</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the covariance between 
               matching elements of two std::vectors.
         </description>
      </component>
      
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      <component>
         <name>r_squared</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the R squared coefficient between 
               matching elements of two std::vectors.
         </description>
      </component>
      
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      <component>
         <name>mean_squared_error</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the mean squared error between 
               matching elements of two std::vectors.
         </description>
      </component>
      
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      <component>
         <name>random_subset_selector</name>
         <file>dlib/statistics.h</file>
         <spec_file>dlib/statistics/random_subset_selector_abstract.h</spec_file>
         <description>
               This object is a tool to help you select a random subset of a large body of data.  
               In particular, it is useful when the body of data is too large to fit into memory.
         </description>

                                 
      </component>
      
      
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      <component>
         <name>randomly_subsample</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/random_subset_selector_abstract.h</spec_file>
         <description>
            This is a set of convenience functions for 
            creating <a href="#random_subset_selector">random subsets</a> of data.
         </description>
                                 
      </component>
      
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      <component>
         <name>randomly_sample_image_features</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/image_feature_sampling_abstract.h</spec_file>
         <description>
            Given a feature extractor such as the <a href="imaging.html#hog_image">hog_image</a>,
            this routine selects a random subsample of local image feature vectors 
            from a set of images.  
         </description>
                                 
      </component>
      
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      <component>
         <name>hsort_array</name>
         <file>dlib/sort.h</file>
         <spec_file link="true">dlib/sort.h</spec_file>
         <description>
            hsort_array is an implementation of the heapsort algorithm.  It will sort anything that has an 
            array like operator[] interface.  
         </description>
                                 
      </component>
      
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      <component>
         <name>put_in_range</name>
         <file>dlib/algs.h</file>
         <spec_file link="true">dlib/algs.h</spec_file>
         <description>
            This is a simple function that takes a range and a value and returns the given
            value if it is within the range.  If it isn't in the range then it returns the  
            end of range value that is closest.
         </description>
                                 
      </component>
               
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      <component>
         <name>isort_array</name>
         <file>dlib/sort.h</file>
         <spec_file link="true">dlib/sort.h</spec_file>
         <description>
            isort_array is an implementation of the insertion sort algorithm.  It will sort anything that has an 
            array like operator[] interface.   
         </description>
                                 
      </component>
               
      
   <!-- ************************************************************************* -->
      
      <component>
         <name>qsort_array</name>
         <file>dlib/sort.h</file>
         <spec_file link="true">dlib/sort.h</spec_file>
         <description>
            qsort_array is an implementation of the QuickSort algorithm.  It will sort anything that has an array like 
            operator[] interface.  If the quick sort becomes unstable then it switches to a heap sort.  This 
            way sorting is guaranteed to take at most N*log(N) time.
         </description>
                                 
      </component>
            
      
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      <component>
         <name>md5</name>
         <file>dlib/md5.h</file>
         <spec_file>dlib/md5/md5_kernel_abstract.h</spec_file>
         <description>
            This is an implementation of The MD5 Message-Digest Algorithm as described in rfc1321.   
         </description>
                                 
      </component>
            
      
      
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      <component>
         <name>median</name>
         <file>dlib/algs.h</file>
         <spec_file link="true">dlib/algs.h</spec_file>
         <description>
            This function takes three parameters and finds the median of the three.  The median is swapped into
            the first parameter and the first parameter ends up in one of the other two, unless the first parameter was
            the median to begin with of course. 
         </description>
                                 
      </component>
            
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      <component>
         <name>square_root</name>
         <file>dlib/algs.h</file>
         <spec_file link="true">dlib/algs.h</spec_file>
         <description>        
            square_root is a function which takes an unsigned long and returns the square root of it or
            if the root is not an integer then it is rounded up to the next integer.
         </description>
                                 
      </component>

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      <component>
         <name>edge</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#graph">graph</a> object and a 
            pair of indices.  It returns a reference to the edge object between the two nodes
            with the given indices. 
         </description>
                                 
      </component>

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      <component>
         <name>is_join_tree</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#graph">graph</a> objects and
            checks if the second of the two graphs is a valid join tree (aka tree decomposition)
            of the first graph.
         </description>
                                 
      </component>

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      <component>
         <name>create_join_tree</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#graph">graph</a> object and
            creates a join tree for that graph.  Or in other words, this function finds a 
            tree decomposition of the given graph.
         </description>
                                 
      </component>

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      <component>
         <name>graph_contains_directed_cycle</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function checks a <a href="containers.html#directed_graph">directed_graph</a> for directed cycles.
         </description>
                                 
      </component>

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      <component>
         <name>set_intersection</name>
         <file>dlib/set_utils.h</file>
         <spec_file link="true">dlib/set_utils/set_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#set">set</a> objects and 
            gives you their intersection.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>set_union</name>
         <file>dlib/set_utils.h</file>
         <spec_file link="true">dlib/set_utils/set_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#set">set</a> objects and 
            gives you their union.
         </description>
                                 
      </component>

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      <component>
         <name>set_difference</name>
         <file>dlib/set_utils.h</file>
         <spec_file link="true">dlib/set_utils/set_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#set">set</a> objects and 
            gives you their difference.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>set_intersection_size</name>
         <file>dlib/set_utils.h</file>
         <spec_file link="true">dlib/set_utils/set_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#set">set</a> objects and tells you
            how many items they have in common.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>triangulate_graph_and_find_cliques</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#graph">graph</a> and
            turns it into a chordal graph.  It also returns a 
            <a href="containers.html#set">set</a> that contains
            all the cliques present in the chordal graph.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>create_moral_graph</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#directed_graph">directed_graph</a>
            and returns the moralized version of the graph in the form of a 
            <a href="containers.html#graph">graph</a> object.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>graph_contains_length_one_cycle</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#graph">graph</a>
            or <a href="containers.html#directed_graph">directed_graph</a> object and
            returns true if and only if the graph contains a node that has an edge that
            links back to itself.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>find_connected_nodes</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a node from a <a href="containers.html#graph">graph</a>
            or <a href="containers.html#directed_graph">directed_graph</a> object and a 
            <a href="containers.html#set">set</a> of unsigned longs.  It finds all the
            nodes in the given graph that are connected to the given node by an 
            undirected path and returns them in the set (also note that the
            original query node is also returned in this set).
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>graph_is_connected</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#graph">graph</a> or
            <a href="containers.html#directed_graph">directed_graph</a> object and
            determines if the graph is connected.  That is, it returns true if and only if
            there is an undirected path between any two nodes in the given graph.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>is_clique</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#graph">graph</a> and a 
            <a href="containers.html#set">set</a> of node index values and checks 
            if the specified set of nodes is a clique in the graph.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>copy_graph_structure</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#graph">graph</a> or 
            <a href="containers.html#directed_graph">directed_graph</a> and copies  
            its structure to another graph or directed_graph object.  The only 
            restriction is that you can't copy the structure of a graph into a
            directed_graph.  The three other possible combinations are allowed
            however.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>is_maximal_clique</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function takes a <a href="containers.html#graph">graph</a> and a 
            <a href="containers.html#set">set</a> of node index values and checks 
            if the specified set of nodes is a maximal clique in the graph.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>nearest_point</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> and a 
            <a href="#point">point</a> and returns the point in the given 
            rectangle that is nearest to the given point.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>distance_to_rect_edge</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> and a 
            <a href="#point">point</a> and returns the Manhattan distance between
            the rectangle's edge and the point.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>move_rect</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> and moves
            it so that it's upper left corner occupies the given location.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>resize_rect_height</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> and 
            returns a new rectangle with the given height but otherwise with the 
            same edge points as the original rectangle. 
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>resize_rect_width</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> and 
            returns a new rectangle with the given width but otherwise with the 
            same edge points as the original rectangle. 
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>resize_rect</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> and 
            returns a new rectangle with the given size but with the same upper
            left corner as the original rectangle.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>translate_rect</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> and moves
            it by a given number of units along the x and y axis relative to
            where it was before the move.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>quantum_register</name>
         <file>dlib/quantum_computing.h</file>
         <spec_file link="true">dlib/quantum_computing/quantum_computing_abstract.h</spec_file>
         <description>        
                This object represents a set of quantum bits.  It can be used
                with the quantum <a href="#gate">gate</a> object to simulate
                quantum algorithms.
         </description>

         <examples>
            <example>quantum_computing_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>gate</name>
         <file>dlib/quantum_computing.h</file>
         <spec_file link="true">dlib/quantum_computing/quantum_computing_abstract.h</spec_file>
         <description>        
            This object represents a quantum gate that operates on a 
            <a href="#quantum_register">quantum_register</a>.   
         </description>
         <examples>
            <example>quantum_computing_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>center</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            Returns the center point of a <a href="#rectangle">rectangle</a>.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>centered_rect</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            There are various overloads of this function but the basic idea is
            that it returns a <a href="#rectangle">rectangle</a> with a given 
            width and height and centered about a given point.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>shrink_rect</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> object, 
            shrinks its borders by a given amount, and returns the result.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>grow_rect</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This function takes a <a href="#rectangle">rectangle</a> object, 
            grows its borders by a given amount, and returns the result.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>rotate_point</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/vector_abstract.h</spec_file>
         <description>        
            This is a function that rotates a 2D <a href="#vector">vector</a> or 
            <a href="#point">point</a> object about a given point.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>point_rotator</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/vector_abstract.h</spec_file>
         <description>        
            This is an object that rotates a 2D <a href="#vector">vector</a> or 
            <a href="#point">point</a> object about the origin.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>get_rect</name>
         <file>dlib/geometry.h</file>
         <spec_file link="true">dlib/geometry/rectangle_abstract.h</spec_file>
         <description>        
            This is a simple template function that returns a rectangle 
            representing the size of a 2D container (e.g. <a href="containers.html#matrix">matrix</a> or
            <a href="containers.html#array2d">array2d</a>). 
         </description>
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>graph_contains_undirected_cycle</name>
         <file>dlib/graph_utils.h</file>
         <spec_file link="true">dlib/graph_utils/graph_utils_abstract.h</spec_file>
         <description>        
            This function checks a <a href="containers.html#directed_graph">directed_graph</a> for undirected cycles.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

   </components>

   <!-- ************************************************************************* -->


</doc>

